Case Study 3

Michael Daniel Bigler and Liam Arthur Phan

2023-04-29

Packages

library(psych)
library(corrplot)
library(ggplot2)
library(car)
library(naniar)
library(REdaS)
library(zoo)
library(foreign) 
library(lavaan)
library(lavaanPlot)
library(ggcorrplot)
library(lares)
library(MVN)
library(dplyr)
library(knitr)

Data

df <- read.csv2('Case Study III_Structural Equation Modeling.csv', na.strings = '999', sep = ',')
df <- df[, c(1:23, 25:36)]

DT::datatable(df)

Dimensions

dim_before_na <- dim(df)
dim_before_na
## [1] 553  35

Summary Statistics

DT::datatable(describe(df))

Missing Analysis

gg_miss_var(df, show_pct = TRUE)

This is not too bad, we can see that SAT_3 is the one with the most NA values, up to 7%.

naniar::vis_miss(df)

If we look at this plot though we see that the missing values are in a lot of the observations. Therefore, we will to handle the Confirmatory Analysis with a method for replacing those missing values.

Dimensions after listwise deletion

dim_after_na <- dim(na.omit(df))
dim_after_na
## [1] 309  35
na_remove_count <- dim_after_na - dim_before_na
na_remove_count[1] <- abs(na_remove_count[1])

Thus, we remove a lot of observations with listwise deletion, up to 244

# We do list-wise deletion as ask by the TA
df_listwise <- na.omit(df)

Assumptions for EFA

From Assistant Please only consider variables image1 to image22, and use listwise deletion to handle missing data before starting exploratory factor analysis.

Basic Assumptions

df_1 <- df_listwise[,1:22]

Normality - Shapiro Wilk’s test

apply(df_1, 2, shapiro.test)
## $Im1
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.92373, p-value = 1.851e-11
## 
## 
## $Im2
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.92499, p-value = 2.411e-11
## 
## 
## $Im3
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.92371, p-value = 1.844e-11
## 
## 
## $Im4
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.92014, p-value = 8.873e-12
## 
## 
## $Im5
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.91233, p-value = 1.921e-12
## 
## 
## $Im6
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.82674, p-value < 2.2e-16
## 
## 
## $Im7
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.84612, p-value < 2.2e-16
## 
## 
## $Im8
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.80379, p-value < 2.2e-16
## 
## 
## $Im9
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.92158, p-value = 1.187e-11
## 
## 
## $Im10
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.7981, p-value < 2.2e-16
## 
## 
## $Im11
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.85448, p-value < 2.2e-16
## 
## 
## $Im12
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.86819, p-value = 1.38e-15
## 
## 
## $Im13
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.89122, p-value = 4.669e-14
## 
## 
## $Im14
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.79446, p-value < 2.2e-16
## 
## 
## $Im15
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.91767, p-value = 5.414e-12
## 
## 
## $Im16
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.90488, p-value = 4.853e-13
## 
## 
## $Im17
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.90676, p-value = 6.818e-13
## 
## 
## $Im18
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.93189, p-value = 1.081e-10
## 
## 
## $Im19
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.90393, p-value = 4.097e-13
## 
## 
## $Im20
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.92824, p-value = 4.834e-11
## 
## 
## $Im21
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.89126, p-value = 4.7e-14
## 
## 
## $Im22
## 
##  Shapiro-Wilk normality test
## 
## data:  newX[, i]
## W = 0.94596, p-value = 3.211e-09

We reject null-hypothesis for all variables and thus don’t accept normality of the data.

Multivariate normality - Mardia’s Multivariate Normality Test

To say the data are multivariate normal:

• z-kurtosis < 5 (Bentler, 2006) and the P-value should be ≥ 0.05. • The plot should also form a straight line (Arifin, 2015).

MVN::mvn(df_1, mvnTest = "mardia", multivariatePlot = "qq", desc = FALSE)

## $multivariateNormality
##              Test        Statistic p value Result
## 1 Mardia Skewness 5693.58260956909       0     NO
## 2 Mardia Kurtosis 48.5688465675536       0     NO
## 3             MVN             <NA>    <NA>     NO
## 
## $univariateNormality
##                Test  Variable Statistic   p value Normality
## 1  Anderson-Darling    Im1       9.2497  <0.001      NO    
## 2  Anderson-Darling    Im2       9.0183  <0.001      NO    
## 3  Anderson-Darling    Im3       9.0339  <0.001      NO    
## 4  Anderson-Darling    Im4       9.6661  <0.001      NO    
## 5  Anderson-Darling    Im5      10.8951  <0.001      NO    
## 6  Anderson-Darling    Im6      18.6917  <0.001      NO    
## 7  Anderson-Darling    Im7      17.3319  <0.001      NO    
## 8  Anderson-Darling    Im8      21.2033  <0.001      NO    
## 9  Anderson-Darling    Im9       9.2733  <0.001      NO    
## 10 Anderson-Darling   Im10      22.1406  <0.001      NO    
## 11 Anderson-Darling   Im11      16.4825  <0.001      NO    
## 12 Anderson-Darling   Im12      14.7775  <0.001      NO    
## 13 Anderson-Darling   Im13      12.4484  <0.001      NO    
## 14 Anderson-Darling   Im14      21.7494  <0.001      NO    
## 15 Anderson-Darling   Im15      10.1132  <0.001      NO    
## 16 Anderson-Darling   Im16      11.4400  <0.001      NO    
## 17 Anderson-Darling   Im17      10.7737  <0.001      NO    
## 18 Anderson-Darling   Im18       8.0021  <0.001      NO    
## 19 Anderson-Darling   Im19      12.4287  <0.001      NO    
## 20 Anderson-Darling   Im20       7.9905  <0.001      NO    
## 21 Anderson-Darling   Im21      13.1862  <0.001      NO    
## 22 Anderson-Darling   Im22       6.1723  <0.001      NO

The data are not normally distributed at multivariate level. Our extraction method PAF can deal with this non-normality.

Multicolinearity

# Correlation Values Matrix
M <- cor(df_1)

# P-Value
p.mat <- cor_pmat(df_1)
Correlation Plot
# Correlation Plot
ggcorrplot(M, hc.order = TRUE, type = "lower", lab = TRUE, p.mat = p.mat, sig.level=0.05, lab_size = 2, tl.cex = 10,outline.col = "white", ggtheme = ggplot2::theme_minimal(), colors = c("#823038", "white", "#2596be")) 

Correlation Ranking
# Ranked Cross-Correlations
corr_cross(df_1, # name of dataset
  max_pvalue = 0.05, # display only significant correlations (at 5% level)
  top = 9 # display top 10 couples of variables (by correlation coefficient)
)

As we can see, We have some multicolinearity amongst the variables, at least 6 variables can be considered with high-colinearity. Im3+Im4, Im1+Im2m, Im6+Im7, Im4+Im5, Im8+Im10 and Im8+Im14.

A guide to appropriate use of Correlation coefficient in medical research

Factors Analysis Assumptions

Kaiser-Meyer-Olkin test (KMO)

KMO: Find the Kaiser, Meyer, Olkin Measure of Sampling Adequacy

KMO Index

KMOTEST <- KMO(M)
sort(KMOTEST$MSAi)
##       Im6      Im10      Im14       Im2       Im1       Im7      Im20      Im17 
## 0.7791619 0.8192843 0.8206186 0.8275596 0.8316756 0.8342908 0.8369658 0.8459668 
##      Im18       Im4       Im3      Im13      Im12      Im22      Im21      Im11 
## 0.8479170 0.8623498 0.8647696 0.8749019 0.8763560 0.8850423 0.8930068 0.9101259 
##      Im16       Im8       Im9      Im19      Im15       Im5 
## 0.9168866 0.9231784 0.9240378 0.9432565 0.9558911 0.9616355

Most KMO Index are Middling, Meritorious or even Marvelous. Im6 is the lowest KMO index being Middling.

KMO Overall Measure of sampling adequacy

KMOTEST$MSA
## [1] 0.8739058

With 0.87, sampling adequacy is very high.

Bartlett’s Test of Sphericity

cortest.bartlett(df_1)
## $chisq
## [1] 5268.134
## 
## $p.value
## [1] 0
## 
## $df
## [1] 231

EFA can be done as the test indicates a p-value under 0 (P-value < 0) and thus can reject the null hypothesis (Identity Matrix).

Exploratory Factor analysis

Determine the number of factors

  1. Kaiser’s eigenvalue > 1 rule.
  2. Cattell’s scree test.
  3. Parallel analysis.
  4. Very simple structure (VSS).
  5. Velicer’s minimum average partial (MAP).

Kaiser’s eigevalue > 1 rule

Factors with eigenvalues > 1 are retained. Eigenvalue can be interpreted as the proportion of the information in a factor. The cut-off of 1 means the factor contains information = 1 item. Thus it is not worthwhile keeping factor with information < 1 item.

fa_result <- fa(df_1, rotate = "varimax", fm = "pa")

factors_kaiser <- sum(fa_result$e.values>1)

print(paste("Kaiser-Criterion:", factors_kaiser,"Factors"))
## [1] "Kaiser-Criterion: 6 Factors"

According to the Kaiser-Criterion, we would use 6 factors.

Catell’s scree test

We can do a factor analysis using rotation varimax

fa_result <- fa(df_1, rotate = "varimax", fm = "pa")
n_factors <- length(fa_result$e.values)
scree <- data.frame(Factor_n =  as.factor(1:n_factors), Eigenvalue = fa_result$e.values)

ggplot(scree, aes(x = Factor_n, y = Eigenvalue, group = 1)) +
  geom_point() + geom_line() +
  xlab("Number of factors") +
  ylab("Initial eigenvalue") +
  labs( title = "Scree Plot",
        subtitle = "(Based on the unreduced correlation matrix)") +
  geom_hline(yintercept = 1, color="#2596be") + theme_minimal() 

We would say 6 factors (above blue line of eigenvalue > 1)

Parallel analysis

parallel <- fa.parallel(df_1, fm = "pa", fa = "fa")

## Parallel analysis suggests that the number of factors =  7  and the number of components =  NA
print(parallel)
## Call: fa.parallel(x = df_1, fm = "pa", fa = "fa")
## Parallel analysis suggests that the number of factors =  7  and the number of components =  NA 
## 
##  Eigen Values of 
## 
##  eigen values of factors
##  [1]  8.52  1.78  0.92  0.76  0.65  0.58  0.20  0.13 -0.09 -0.12 -0.21 -0.25
## [13] -0.29 -0.34 -0.37 -0.40 -0.42 -0.42 -0.47 -0.51 -0.54 -0.60
## 
##  eigen values of simulated factors
##  [1]  0.61  0.45  0.39  0.32  0.27  0.22  0.17  0.13  0.09  0.06  0.02 -0.02
## [13] -0.05 -0.08 -0.12 -0.15 -0.19 -0.22 -0.26 -0.29 -0.34 -0.39
## 
##  eigen values of components 
##  [1] 9.11 2.46 1.58 1.36 1.26 1.14 0.79 0.73 0.56 0.46 0.36 0.33 0.30 0.28 0.25
## [16] 0.22 0.19 0.18 0.14 0.11 0.10 0.08
## 
##  eigen values of simulated components
## [1] NA

As we can see in parallel analysis, it also suggest 6 factors, nevertheless, factors up to 7 or 8 can also be considered.

Very simple structure (VSS) criterion and Velicer’s minimum average partial (MAP) criterion

vss(df_1, rotate = "varimax", fm = "pa")

## 
## Very Simple Structure
## Call: vss(x = df_1, rotate = "varimax", fm = "pa")
## VSS complexity 1 achieves a maximimum of 0.84  with  1  factors
## VSS complexity 2 achieves a maximimum of 0.9  with  2  factors
## 
## The Velicer MAP achieves a minimum of 0.04  with  8  factors 
## BIC achieves a minimum of  -334.88  with  8  factors
## Sample Size adjusted BIC achieves a minimum of  -71.64  with  8  factors
## 
## Statistics by number of factors 
##   vss1 vss2   map dof chisq     prob sqresid  fit RMSEA  BIC SABIC complex
## 1 0.84 0.00 0.047 209  2756  0.0e+00    16.0 0.84 0.199 1558  2221     1.0
## 2 0.71 0.90 0.044 188  2143  0.0e+00    10.1 0.90 0.183 1065  1661     1.3
## 3 0.62 0.89 0.045 168  1773 1.0e-265     7.6 0.92 0.176  810  1343     1.4
## 4 0.52 0.84 0.047 149  1455 8.0e-213     5.9 0.94 0.168  600  1073     1.7
## 5 0.44 0.76 0.044 131  1139 3.0e-160     4.3 0.96 0.158  388   803     1.9
## 6 0.39 0.63 0.042 114   636  1.9e-73     3.0 0.97 0.122  -18   344     2.1
## 7 0.39 0.60 0.045  98   296  1.1e-21     2.3 0.98 0.081 -266    45     1.9
## 8 0.37 0.54 0.036  83   141  7.5e-05     1.7 0.98 0.047 -335   -72     2.0
##   eChisq  SRMR eCRMS eBIC
## 1   2321 0.127 0.134 1122
## 2   1213 0.092 0.102  135
## 3    837 0.077 0.090 -126
## 4    588 0.064 0.080 -266
## 5    364 0.051 0.067 -387
## 6    157 0.033 0.047 -497
## 7     74 0.023 0.035 -488
## 8     19 0.012 0.019 -456

VSS indicates 1/2 factors (vss1 largest at 1 and 2 factors), while MAP indicates 8 factors (map smallest at 8 factors).

VSS criterion for the number of factors (in R’s psych package)

Extraction Method

Our data are not normally distributed, hence the extraction method of choice is principal axis factoring (PAF), because it does not assume normality of data (Brown, 2015). The rotation method is varimax.

We run EFA by

  1. fixing the number of factors as decided from previous step. 6 or 8 factors are reasonable.
  2. choosing an appropriate extraction method. We use PAF, fm = “pa” (Principal Axis Factoring).
  3. choosing an appropriate rotation method. We use varimax, rotate = “varimax”.

6 Factors

We will compute the loadings with 6 factors and varimax rotation

What we need to look for:

  1. Factor loadings

Multiple threshold exist (as many rules of thumb), in our analysis we will use the standard 0.4 cut-off.

What thresholds should I use for factor loading cut-offs?

  1. Communalities

We use the standard cut-off of 0.5, all above are good.

fa_result <- fa(df_1, nfactors = 6, fm = "pa", rotate = "varimax")

print(fa_result, cut = 0.4, digits = 3)
## Factor Analysis using method =  pa
## Call: fa(r = df_1, nfactors = 6, rotate = "varimax", fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
##        PA5   PA1    PA2   PA4   PA3   PA6    h2    u2  com
## Im1  0.850                                0.841 0.159 1.34
## Im2  0.842                                0.796 0.204 1.25
## Im3        0.831                          0.874 0.126 1.58
## Im4        0.850                          0.893 0.107 1.51
## Im5        0.603                          0.524 0.476 1.98
## Im6                                 0.783 0.708 0.292 1.31
## Im7               0.445             0.739 0.767 0.233 1.75
## Im8               0.725                   0.716 0.284 1.72
## Im9                                 0.480 0.453 0.547 2.80
## Im10              0.821                   0.793 0.207 1.37
## Im11                    0.537             0.461 0.539 2.24
## Im12                    0.796             0.758 0.242 1.42
## Im13                    0.748             0.725 0.275 1.64
## Im14              0.794                   0.760 0.240 1.43
## Im15 0.589                                0.641 0.359 2.86
## Im16 0.468                                0.461 0.539 3.00
## Im17                    0.439       0.416 0.666 0.334 4.39
## Im18                    0.438             0.605 0.395 4.18
## Im19 0.478 0.405                          0.541 0.459 3.38
## Im20                          0.839       0.775 0.225 1.21
## Im21                          0.766       0.680 0.320 1.34
## Im22                          0.798       0.804 0.196 1.56
## 
##                         PA5   PA1   PA2   PA4   PA3   PA6
## SS loadings           2.864 2.782 2.608 2.514 2.395 2.080
## Proportion Var        0.130 0.126 0.119 0.114 0.109 0.095
## Cumulative Var        0.130 0.257 0.375 0.489 0.598 0.693
## Proportion Explained  0.188 0.183 0.171 0.165 0.157 0.136
## Cumulative Proportion 0.188 0.370 0.541 0.706 0.864 1.000
## 
## Mean item complexity =  2.1
## Test of the hypothesis that 6 factors are sufficient.
## 
## df null model =  231  with the objective function =  17.57 with Chi Square =  5268.134
## df of  the model are 114  and the objective function was  2.15 
## 
## The root mean square of the residuals (RMSR) is  0.033 
## The df corrected root mean square of the residuals is  0.047 
## 
## The harmonic n.obs is  309 with the empirical chi square  157.019  with prob <  0.00471 
## The total n.obs was  309  with Likelihood Chi Square =  635.908  with prob <  1.9e-73 
## 
## Tucker Lewis Index of factoring reliability =  0.7871
## RMSEA index =  0.1217  and the 90 % confidence intervals are  0.1128 0.1312
## BIC =  -17.693
## Fit based upon off diagonal values = 0.993
## Measures of factor score adequacy             
##                                                     PA5   PA1   PA2   PA4   PA3
## Correlation of (regression) scores with factors   0.933 0.949 0.930 0.906 0.930
## Multiple R square of scores with factors          0.871 0.900 0.864 0.821 0.865
## Minimum correlation of possible factor scores     0.741 0.800 0.729 0.643 0.731
##                                                     PA6
## Correlation of (regression) scores with factors   0.904
## Multiple R square of scores with factors          0.817
## Minimum correlation of possible factor scores     0.635

Exploratory factor analysis and Cronbach’s alpha Questionnaire Validation Workshop, 10/10/2017, USM Health Campus

1. Factor loadings

We can see that we have 3 cross-loadings, Im7, Im17 and Im19.

Cross-Loadings (Measured with Complexity measure: com > 1):

Im17 > Im19 > Im7 > 1

2. Communalities

On the table, it is column h2

Low Communalities are :

Im9 < Im11 < Im16 < 0.5

Removing Im17 (Lowest Communality and High Complexity)

fa_result <- fa(df_1[!names(df_1) %in% c("Im17")], nfactors = 6, fm = "pa", rotate = "varimax")

print(fa_result, cut = 0.4, digits = 3)
## Factor Analysis using method =  pa
## Call: fa(r = df_1[!names(df_1) %in% c("Im17")], nfactors = 6, rotate = "varimax", 
##     fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
##        PA1   PA5   PA2   PA3   PA4   PA6    h2     u2  com
## Im1  0.855                               0.848 0.1519 1.33
## Im2  0.845                               0.799 0.2007 1.25
## Im3        0.839                         0.884 0.1159 1.55
## Im4        0.871                         0.923 0.0769 1.46
## Im5        0.605                         0.526 0.4743 1.97
## Im6                                0.860 0.795 0.2050 1.15
## Im7                                0.792 0.786 0.2142 1.51
## Im8              0.673             0.429 0.692 0.3077 1.99
## Im9                                0.471 0.449 0.5511 2.83
## Im10             0.873                   0.878 0.1222 1.32
## Im11                         0.548       0.458 0.5423 2.13
## Im12                         0.835       0.805 0.1950 1.33
## Im13                         0.757       0.738 0.2618 1.63
## Im14             0.830                   0.818 0.1820 1.40
## Im15 0.593                               0.643 0.3570 2.80
## Im16 0.468                               0.461 0.5390 3.00
## Im18                                     0.421 0.5791 4.38
## Im19 0.480 0.403                         0.538 0.4623 3.36
## Im20                   0.836             0.770 0.2296 1.21
## Im21                   0.773             0.684 0.3157 1.30
## Im22                   0.800             0.804 0.1959 1.55
## 
##                         PA1   PA5   PA2   PA3   PA4   PA6
## SS loadings           2.778 2.690 2.447 2.382 2.367 2.057
## Proportion Var        0.132 0.128 0.117 0.113 0.113 0.098
## Cumulative Var        0.132 0.260 0.377 0.490 0.603 0.701
## Proportion Explained  0.189 0.183 0.166 0.162 0.161 0.140
## Cumulative Proportion 0.189 0.371 0.538 0.699 0.860 1.000
## 
## Mean item complexity =  1.9
## Test of the hypothesis that 6 factors are sufficient.
## 
## df null model =  210  with the objective function =  16.063 with Chi Square =  4821.666
## df of  the model are 99  and the objective function was  1.019 
## 
## The root mean square of the residuals (RMSR) is  0.025 
## The df corrected root mean square of the residuals is  0.036 
## 
## The harmonic n.obs is  309 with the empirical chi square  81.338  with prob <  0.902 
## The total n.obs was  309  with Likelihood Chi Square =  301.667  with prob <  2.4e-22 
## 
## Tucker Lewis Index of factoring reliability =  0.9055
## RMSEA index =  0.0813  and the 90 % confidence intervals are  0.0711 0.0921
## BIC =  -265.934
## Fit based upon off diagonal values = 0.996
## Measures of factor score adequacy             
##                                                     PA1   PA5   PA2   PA3   PA4
## Correlation of (regression) scores with factors   0.935 0.959 0.943 0.931 0.915
## Multiple R square of scores with factors          0.875 0.920 0.889 0.866 0.837
## Minimum correlation of possible factor scores     0.750 0.841 0.777 0.733 0.673
##                                                     PA6
## Correlation of (regression) scores with factors   0.924
## Multiple R square of scores with factors          0.854
## Minimum correlation of possible factor scores     0.709

Removing Im18 (Lowest loadings and High Complexity)

fa_result <- fa(df_1[!names(df_1) %in% c("Im17","Im18")], nfactors = 6, fm = "pa", rotate = "varimax")

print(fa_result, cut = 0.4, digits = 3)
## Factor Analysis using method =  pa
## Call: fa(r = df_1[!names(df_1) %in% c("Im17", "Im18")], nfactors = 6, 
##     rotate = "varimax", fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
##        PA1   PA5   PA2   PA3   PA4   PA6    h2     u2  com
## Im1  0.859                               0.853 0.1475 1.32
## Im2  0.846                               0.798 0.2016 1.24
## Im3        0.844                         0.895 0.1053 1.55
## Im4        0.873                         0.928 0.0722 1.47
## Im5        0.599                         0.520 0.4797 2.00
## Im6                                0.859 0.787 0.2125 1.14
## Im7                                0.810 0.799 0.2007 1.44
## Im8              0.652             0.448 0.685 0.3146 2.12
## Im9                                0.461 0.427 0.5731 2.87
## Im10             0.877                   0.890 0.1099 1.33
## Im11                         0.549       0.458 0.5420 2.14
## Im12                         0.853       0.835 0.1654 1.31
## Im13                         0.743       0.723 0.2772 1.67
## Im14             0.829                   0.823 0.1771 1.42
## Im15 0.597                               0.644 0.3563 2.76
## Im16 0.470                               0.460 0.5402 2.98
## Im19 0.482                               0.535 0.4651 3.33
## Im20                   0.838             0.772 0.2276 1.20
## Im21                   0.773             0.681 0.3192 1.29
## Im22                   0.802             0.805 0.1952 1.54
## 
##                         PA1   PA5   PA2   PA3   PA4   PA6
## SS loadings           2.738 2.565 2.379 2.353 2.223 2.059
## Proportion Var        0.137 0.128 0.119 0.118 0.111 0.103
## Cumulative Var        0.137 0.265 0.384 0.502 0.613 0.716
## Proportion Explained  0.191 0.179 0.166 0.164 0.155 0.144
## Cumulative Proportion 0.191 0.370 0.537 0.701 0.856 1.000
## 
## Mean item complexity =  1.8
## Test of the hypothesis that 6 factors are sufficient.
## 
## df null model =  190  with the objective function =  15.509 with Chi Square =  4660.488
## df of  the model are 85  and the objective function was  0.921 
## 
## The root mean square of the residuals (RMSR) is  0.025 
## The df corrected root mean square of the residuals is  0.037 
## 
## The harmonic n.obs is  309 with the empirical chi square  72.686  with prob <  0.827 
## The total n.obs was  309  with Likelihood Chi Square =  273.088  with prob <  1.36e-21 
## 
## Tucker Lewis Index of factoring reliability =  0.9046
## RMSEA index =  0.0846  and the 90 % confidence intervals are  0.0736 0.0961
## BIC =  -214.246
## Fit based upon off diagonal values = 0.996
## Measures of factor score adequacy             
##                                                     PA1   PA5   PA2   PA3   PA4
## Correlation of (regression) scores with factors   0.937 0.963 0.946 0.931 0.921
## Multiple R square of scores with factors          0.877 0.927 0.894 0.868 0.847
## Minimum correlation of possible factor scores     0.755 0.855 0.789 0.735 0.695
##                                                     PA6
## Correlation of (regression) scores with factors   0.926
## Multiple R square of scores with factors          0.858
## Minimum correlation of possible factor scores     0.717

Removing Im8 (Cross-loadings (High Complexity))

fa_result <- fa(df_1[!names(df_1) %in% c("Im17","Im18","Im8")], nfactors = 6, fm = "pa", rotate = "varimax")

print(fa_result, cut = 0.4, digits = 3)
## Factor Analysis using method =  pa
## Call: fa(r = df_1[!names(df_1) %in% c("Im17", "Im18", "Im8")], nfactors = 6, 
##     rotate = "varimax", fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
##        PA1   PA5   PA3   PA4   PA2   PA6    h2     u2  com
## Im1  0.859                               0.853 0.1471 1.32
## Im2  0.844                               0.796 0.2045 1.24
## Im3        0.845                         0.893 0.1070 1.54
## Im4        0.875                         0.928 0.0721 1.45
## Im5        0.598                         0.519 0.4808 2.00
## Im6                          0.884       0.817 0.1827 1.10
## Im7                          0.806       0.768 0.2316 1.38
## Im9                          0.464       0.427 0.5730 2.80
## Im10                               0.882 0.924 0.0763 1.39
## Im11                   0.553             0.455 0.5449 2.08
## Im12                   0.861             0.841 0.1592 1.28
## Im13                   0.740             0.717 0.2833 1.67
## Im14                               0.811 0.824 0.1761 1.54
## Im15 0.599                               0.645 0.3550 2.72
## Im16 0.472                               0.454 0.5461 2.89
## Im19 0.483 0.405                         0.534 0.4664 3.27
## Im20             0.836                   0.770 0.2303 1.21
## Im21             0.774                   0.682 0.3183 1.29
## Im22             0.803                   0.807 0.1933 1.54
## 
##                         PA1   PA5   PA3   PA4   PA2   PA6
## SS loadings           2.730 2.578 2.347 2.243 1.942 1.812
## Proportion Var        0.144 0.136 0.124 0.118 0.102 0.095
## Cumulative Var        0.144 0.279 0.403 0.521 0.623 0.719
## Proportion Explained  0.200 0.189 0.172 0.164 0.142 0.133
## Cumulative Proportion 0.200 0.389 0.561 0.725 0.867 1.000
## 
## Mean item complexity =  1.8
## Test of the hypothesis that 6 factors are sufficient.
## 
## df null model =  171  with the objective function =  14.415 with Chi Square =  4336.563
## df of  the model are 72  and the objective function was  0.819 
## 
## The root mean square of the residuals (RMSR) is  0.025 
## The df corrected root mean square of the residuals is  0.039 
## 
## The harmonic n.obs is  309 with the empirical chi square  67.363  with prob <  0.633 
## The total n.obs was  309  with Likelihood Chi Square =  243.141  with prob <  2.01e-20 
## 
## Tucker Lewis Index of factoring reliability =  0.9011
## RMSEA index =  0.0876  and the 90 % confidence intervals are  0.0758 0.1001
## BIC =  -169.66
## Fit based upon off diagonal values = 0.996
## Measures of factor score adequacy             
##                                                     PA1   PA5   PA3   PA4   PA2
## Correlation of (regression) scores with factors   0.936 0.963 0.931 0.923 0.930
## Multiple R square of scores with factors          0.877 0.928 0.867 0.852 0.866
## Minimum correlation of possible factor scores     0.754 0.855 0.735 0.703 0.731
##                                                     PA6
## Correlation of (regression) scores with factors   0.953
## Multiple R square of scores with factors          0.908
## Minimum correlation of possible factor scores     0.816

Removing Im19 (Low Communality and High Complexity)

fa_result <- fa(df_1[!names(df_1) %in% c("Im17","Im18","Im8","Im19")], nfactors = 6, fm = "pa", rotate = "varimax")

print(fa_result, cut = 0.4, digits = 3)
## Factor Analysis using method =  pa
## Call: fa(r = df_1[!names(df_1) %in% c("Im17", "Im18", "Im8", "Im19")], 
##     nfactors = 6, rotate = "varimax", fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
##        PA5   PA4   PA3   PA1   PA2   PA6    h2     u2  com
## Im1        0.876                         0.894 0.1059 1.34
## Im2        0.863                         0.839 0.1610 1.26
## Im3  0.843                               0.886 0.1137 1.53
## Im4  0.884                               0.938 0.0620 1.43
## Im5  0.615                               0.538 0.4624 1.93
## Im6                          0.886       0.820 0.1796 1.09
## Im7                          0.805       0.765 0.2346 1.37
## Im9                          0.464       0.428 0.5717 2.79
## Im10                               0.892 0.937 0.0628 1.38
## Im11                   0.553             0.454 0.5462 2.06
## Im12                   0.866             0.843 0.1568 1.26
## Im13                   0.743             0.715 0.2854 1.64
## Im14                               0.812 0.822 0.1779 1.53
## Im15       0.560                         0.623 0.3767 3.06
## Im16       0.405                         0.396 0.6043 3.39
## Im20             0.844                   0.779 0.2213 1.19
## Im21             0.773                   0.679 0.3214 1.28
## Im22             0.801                   0.805 0.1948 1.54
## 
##                         PA5   PA4   PA3   PA1   PA2   PA6
## SS loadings           2.460 2.387 2.343 2.248 1.932 1.792
## Proportion Var        0.137 0.133 0.130 0.125 0.107 0.100
## Cumulative Var        0.137 0.269 0.399 0.524 0.632 0.731
## Proportion Explained  0.187 0.181 0.178 0.171 0.147 0.136
## Cumulative Proportion 0.187 0.368 0.546 0.717 0.864 1.000
## 
## Mean item complexity =  1.7
## Test of the hypothesis that 6 factors are sufficient.
## 
## df null model =  153  with the objective function =  13.52 with Chi Square =  4071.795
## df of  the model are 60  and the objective function was  0.48 
## 
## The root mean square of the residuals (RMSR) is  0.018 
## The df corrected root mean square of the residuals is  0.029 
## 
## The harmonic n.obs is  309 with the empirical chi square  32.204  with prob <  0.999 
## The total n.obs was  309  with Likelihood Chi Square =  142.579  with prob <  1.12e-08 
## 
## Tucker Lewis Index of factoring reliability =  0.9455
## RMSEA index =  0.0667  and the 90 % confidence intervals are  0.0528 0.0811
## BIC =  -201.422
## Fit based upon off diagonal values = 0.998
## Measures of factor score adequacy             
##                                                     PA5   PA4   PA3   PA1   PA2
## Correlation of (regression) scores with factors   0.966 0.950 0.933 0.925 0.931
## Multiple R square of scores with factors          0.933 0.902 0.870 0.856 0.867
## Minimum correlation of possible factor scores     0.867 0.804 0.740 0.712 0.734
##                                                     PA6
## Correlation of (regression) scores with factors   0.958
## Multiple R square of scores with factors          0.919
## Minimum correlation of possible factor scores     0.837

6 Factors - Conclusion

We removed Im17, Im18, Im8 and Im9 until achieving clear loadings separation.

fa.diagram(fa_result, sort = TRUE, adj = 1, rsize = 4, e.size = 0.07, main = "Factors Analysis with 6 factors", digits = 2, l.cex = 1)

Most Factors have good loadings (at least 2 above 0.7), while PA6 has only 2 variables loaded.

8 Factors

We will redo the same analysis with 8 factors this time and using varimax rotation as well.

fa_result <- fa(df_1, nfactors = 8, fm = "pa", rotate = "varimax")

print(fa_result, cut = 0.4, digits = 3)
## Factor Analysis using method =  pa
## Call: fa(r = df_1, nfactors = 8, rotate = "varimax", fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
##        PA3   PA1   PA2   PA6   PA7   PA5    PA4   PA8    h2     u2  com
## Im1                                0.879              0.949 0.0505 1.49
## Im2                                0.822              0.832 0.1679 1.50
## Im3        0.803                                      0.877 0.1232 1.81
## Im4        0.858                                      0.939 0.0606 1.61
## Im5        0.620                                      0.568 0.4319 2.09
## Im6                          0.845                    0.773 0.2268 1.17
## Im7                          0.819                    0.808 0.1924 1.42
## Im8              0.619       0.474                    0.700 0.2995 2.45
## Im9                          0.447                    0.444 0.5557 3.44
## Im10             0.870                                0.887 0.1126 1.37
## Im11                   0.552                          0.467 0.5327 2.18
## Im12                   0.851                          0.841 0.1588 1.35
## Im13                   0.724                          0.730 0.2703 1.88
## Im14             0.850                                0.860 0.1396 1.41
## Im15                               0.442        0.423 0.668 0.3322 4.90
## Im16                                            0.732 0.744 0.2559 1.87
## Im17                                      0.830       0.923 0.0768 1.77
## Im18                                      0.757       0.785 0.2147 1.83
## Im19                                            0.561 0.640 0.3605 3.44
## Im20 0.849                                            0.790 0.2095 1.20
## Im21 0.769                                            0.684 0.3161 1.33
## Im22 0.797                                            0.799 0.2012 1.56
## 
##                         PA3   PA1   PA2   PA6   PA7   PA5   PA4   PA8
## SS loadings           2.441 2.394 2.308 2.244 2.110 2.100 1.718 1.395
## Proportion Var        0.111 0.109 0.105 0.102 0.096 0.095 0.078 0.063
## Cumulative Var        0.111 0.220 0.325 0.427 0.523 0.618 0.696 0.760
## Proportion Explained  0.146 0.143 0.138 0.134 0.126 0.126 0.103 0.083
## Cumulative Proportion 0.146 0.289 0.427 0.562 0.688 0.814 0.917 1.000
## 
## Mean item complexity =  2
## Test of the hypothesis that 8 factors are sufficient.
## 
## df null model =  231  with the objective function =  17.57 with Chi Square =  5268.134
## df of  the model are 83  and the objective function was  0.479 
## 
## The root mean square of the residuals (RMSR) is  0.012 
## The df corrected root mean square of the residuals is  0.019 
## 
## The harmonic n.obs is  309 with the empirical chi square  19.471  with prob <  1 
## The total n.obs was  309  with Likelihood Chi Square =  140.985  with prob <  7.46e-05 
## 
## Tucker Lewis Index of factoring reliability =  0.9674
## RMSEA index =  0.0474  and the 90 % confidence intervals are  0.0337 0.0609
## BIC =  -334.883
## Fit based upon off diagonal values = 0.999
## Measures of factor score adequacy             
##                                                     PA3   PA1   PA2   PA6   PA7
## Correlation of (regression) scores with factors   0.934 0.958 0.947 0.920 0.924
## Multiple R square of scores with factors          0.872 0.917 0.898 0.846 0.854
## Minimum correlation of possible factor scores     0.745 0.834 0.795 0.692 0.709
##                                                     PA5   PA4   PA8
## Correlation of (regression) scores with factors   0.964 0.943 0.838
## Multiple R square of scores with factors          0.930 0.890 0.703
## Minimum correlation of possible factor scores     0.860 0.780 0.405

1. Factor loadings

We can see that we have 2 cross-loadings, Im8 and Im15. Therefore 1 less cross-loadings than 6 Factors Analysis.

Cross-Loadings (Measured with Complexity measure: com > 1):

Im15 > Im8 > 1

2. Communalities

On the table, it is column h2

Low Communalities are :

Im9 < Im11 < 0.5 (same low items communalities than in 6 Factor analysis )

Removing Im15 (Low Communality and High Complexity)

fa_result <- fa(df_1[!names(df_1) %in% c("Im15")], nfactors = 8, fm = "pa", rotate = "varimax")

print(fa_result, cut = 0.4, digits = 3)
## Factor Analysis using method =  pa
## Call: fa(r = df_1[!names(df_1) %in% c("Im15")], nfactors = 8, rotate = "varimax", 
##     fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
##        PA3   PA1   PA2   PA5   PA7   PA6    PA4   PA8    h2     u2  com
## Im1                                0.864              0.935 0.0647 1.55
## Im2                                0.825              0.845 0.1553 1.52
## Im3        0.804                                      0.878 0.1225 1.80
## Im4        0.856                                      0.936 0.0639 1.61
## Im5        0.625                                      0.573 0.4275 2.06
## Im6                          0.850                    0.781 0.2190 1.16
## Im7                          0.817                    0.805 0.1952 1.42
## Im8              0.623       0.472                    0.698 0.3019 2.41
## Im9                          0.444                    0.442 0.5578 3.46
## Im10             0.881                                0.902 0.0979 1.35
## Im11                   0.556                          0.471 0.5294 2.14
## Im12                   0.852                          0.841 0.1587 1.34
## Im13                   0.722                          0.725 0.2749 1.88
## Im14             0.837                                0.844 0.1564 1.43
## Im16                                            0.675 0.677 0.3234 2.11
## Im17                                      0.826       0.912 0.0883 1.75
## Im18                                      0.765       0.794 0.2060 1.80
## Im19                                            0.609 0.686 0.3136 3.02
## Im20 0.854                                            0.796 0.2037 1.19
## Im21 0.770                                            0.683 0.3169 1.32
## Im22 0.796                                            0.795 0.2051 1.55
## 
##                         PA3   PA1   PA2   PA5   PA7   PA6   PA4   PA8
## SS loadings           2.395 2.369 2.303 2.152 2.086 1.841 1.699 1.174
## Proportion Var        0.114 0.113 0.110 0.102 0.099 0.088 0.081 0.056
## Cumulative Var        0.114 0.227 0.336 0.439 0.538 0.626 0.707 0.763
## Proportion Explained  0.150 0.148 0.144 0.134 0.130 0.115 0.106 0.073
## Cumulative Proportion 0.150 0.297 0.441 0.575 0.706 0.821 0.927 1.000
## 
## Mean item complexity =  1.8
## Test of the hypothesis that 8 factors are sufficient.
## 
## df null model =  210  with the objective function =  16.523 with Chi Square =  4959.748
## df of  the model are 70  and the objective function was  0.434 
## 
## The root mean square of the residuals (RMSR) is  0.012 
## The df corrected root mean square of the residuals is  0.02 
## 
## The harmonic n.obs is  309 with the empirical chi square  17.262  with prob <  1 
## The total n.obs was  309  with Likelihood Chi Square =  127.971  with prob <  2.9e-05 
## 
## Tucker Lewis Index of factoring reliability =  0.9627
## RMSEA index =  0.0517  and the 90 % confidence intervals are  0.0374 0.0659
## BIC =  -273.363
## Fit based upon off diagonal values = 0.999
## Measures of factor score adequacy             
##                                                     PA3   PA1   PA2   PA5   PA7
## Correlation of (regression) scores with factors   0.935 0.954 0.949 0.919 0.925
## Multiple R square of scores with factors          0.873 0.911 0.900 0.844 0.856
## Minimum correlation of possible factor scores     0.747 0.821 0.800 0.688 0.712
##                                                     PA6   PA4   PA8
## Correlation of (regression) scores with factors   0.957 0.938 0.806
## Multiple R square of scores with factors          0.917 0.881 0.649
## Minimum correlation of possible factor scores     0.834 0.761 0.299

Removing Im8 (Low Communality and High Complexity)

fa_result <- fa(df_1[!names(df_1) %in% c("Im15","Im8")], nfactors = 8, fm = "pa", rotate = "varimax")

print(fa_result, cut = 0.4, digits = 3)
## Factor Analysis using method =  pa
## Call: fa(r = df_1[!names(df_1) %in% c("Im15", "Im8")], nfactors = 8, 
##     rotate = "varimax", fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
##        PA3   PA1   PA4   PA5   PA6   PA2   PA7   PA8    h2     u2  com
## Im1                          0.898                   0.990 0.0105 1.49
## Im2                          0.792                   0.800 0.1997 1.61
## Im3        0.800                                     0.875 0.1250 1.82
## Im4        0.855                                     0.937 0.0632 1.62
## Im5        0.627                                     0.575 0.4248 2.06
## Im6                    0.885                         0.827 0.1732 1.12
## Im7                    0.804                         0.766 0.2335 1.39
## Im9                    0.443                         0.440 0.5599 3.42
## Im10                               0.914             0.980 0.0201 1.37
## Im11             0.562                               0.468 0.5320 2.07
## Im12             0.857                               0.845 0.1553 1.32
## Im13             0.722                               0.723 0.2773 1.86
## Im14                               0.780             0.789 0.2110 1.64
## Im16                                           0.616 0.601 0.3986 2.35
## Im17                                     0.864       0.969 0.0311 1.66
## Im18                                     0.730       0.752 0.2484 1.92
## Im19                                           0.689 0.758 0.2415 2.40
## Im20 0.852                                           0.794 0.2064 1.19
## Im21 0.770                                           0.683 0.3173 1.32
## Im22 0.798                                           0.798 0.2024 1.55
## 
##                         PA3   PA1   PA4   PA5   PA6   PA2   PA7   PA8
## SS loadings           2.384 2.345 2.169 1.936 1.832 1.788 1.698 1.216
## Proportion Var        0.119 0.117 0.108 0.097 0.092 0.089 0.085 0.061
## Cumulative Var        0.119 0.236 0.345 0.442 0.533 0.623 0.708 0.768
## Proportion Explained  0.155 0.153 0.141 0.126 0.119 0.116 0.110 0.079
## Cumulative Proportion 0.155 0.308 0.449 0.575 0.694 0.810 0.921 1.000
## 
## Mean item complexity =  1.8
## Test of the hypothesis that 8 factors are sufficient.
## 
## df null model =  190  with the objective function =  15.431 with Chi Square =  4636.932
## df of  the model are 58  and the objective function was  0.36 
## 
## The root mean square of the residuals (RMSR) is  0.011 
## The df corrected root mean square of the residuals is  0.021 
## 
## The harmonic n.obs is  309 with the empirical chi square  15.22  with prob <  1 
## The total n.obs was  309  with Likelihood Chi Square =  106.281  with prob <  0.000114 
## 
## Tucker Lewis Index of factoring reliability =  0.9638
## RMSEA index =  0.0518  and the 90 % confidence intervals are  0.036 0.0674
## BIC =  -226.253
## Fit based upon off diagonal values = 0.999
## Measures of factor score adequacy             
##                                                     PA3   PA1   PA4   PA5   PA6
## Correlation of (regression) scores with factors   0.934 0.955 0.922 0.931 0.989
## Multiple R square of scores with factors          0.873 0.911 0.850 0.867 0.978
## Minimum correlation of possible factor scores     0.746 0.823 0.700 0.735 0.955
##                                                     PA2   PA7   PA8
## Correlation of (regression) scores with factors   0.980 0.973 0.818
## Multiple R square of scores with factors          0.961 0.947 0.669
## Minimum correlation of possible factor scores     0.922 0.895 0.338

8 Factors - Conclusion

We removed Im15, Im8 until achieving clear loadings separation. Therefore we removed 2 variables less than 6 Factors Analysis done previously

fa.diagram(fa_result, sort = TRUE, adj = 1, rsize = 4, e.size = 0.07, main = "Factors Analysis with 8 factors", digits = 2, l.cex = 1)

Most Factors have nice loadings (at least 2 above 0.7), but PA8 has 2 variables with only 0.62-0.69 loadings (but close to 0.7).

Deciding between 6 or 8 Factors

fa_result6 <- fa(df_1[!names(df_1) %in% c("Im17","Im18","Im8","Im19")], nfactors = 6, fm = "pa", rotate = "varimax")
fa_result8 <- fa(df_1[!names(df_1) %in% c("Im15","Im8")], nfactors = 8, fm = "pa", rotate = "varimax")

fa_result6
## Factor Analysis using method =  pa
## Call: fa(r = df_1[!names(df_1) %in% c("Im17", "Im18", "Im8", "Im19")], 
##     nfactors = 6, rotate = "varimax", fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
##       PA5  PA4  PA3  PA1  PA2  PA6   h2    u2 com
## Im1  0.19 0.88 0.22 0.18 0.07 0.08 0.89 0.106 1.3
## Im2  0.22 0.86 0.16 0.10 0.07 0.08 0.84 0.161 1.3
## Im3  0.84 0.21 0.22 0.22 0.12 0.14 0.89 0.114 1.5
## Im4  0.88 0.21 0.18 0.20 0.14 0.13 0.94 0.062 1.4
## Im5  0.62 0.24 0.18 0.20 0.08 0.15 0.54 0.462 1.9
## Im6  0.09 0.06 0.07 0.06 0.89 0.12 0.82 0.180 1.1
## Im7  0.06 0.08 0.11 0.11 0.80 0.29 0.77 0.235 1.4
## Im9  0.19 0.13 0.12 0.36 0.46 0.11 0.43 0.572 2.8
## Im10 0.19 0.10 0.05 0.21 0.23 0.89 0.94 0.063 1.4
## Im11 0.18 0.11 0.18 0.55 0.09 0.25 0.45 0.546 2.1
## Im12 0.17 0.15 0.10 0.87 0.10 0.16 0.84 0.157 1.3
## Im13 0.22 0.24 0.19 0.74 0.14 0.07 0.71 0.285 1.6
## Im14 0.17 0.14 0.06 0.20 0.27 0.81 0.82 0.178 1.5
## Im15 0.26 0.56 0.26 0.37 0.17 0.10 0.62 0.377 3.1
## Im16 0.35 0.40 0.13 0.20 0.06 0.22 0.40 0.604 3.4
## Im20 0.14 0.11 0.84 0.17 0.03 0.04 0.78 0.221 1.2
## Im21 0.16 0.18 0.77 0.12 0.08 0.04 0.68 0.321 1.3
## Im22 0.21 0.24 0.80 0.14 0.19 0.06 0.81 0.195 1.5
## 
##                        PA5  PA4  PA3  PA1  PA2  PA6
## SS loadings           2.46 2.39 2.34 2.25 1.93 1.79
## Proportion Var        0.14 0.13 0.13 0.12 0.11 0.10
## Cumulative Var        0.14 0.27 0.40 0.52 0.63 0.73
## Proportion Explained  0.19 0.18 0.18 0.17 0.15 0.14
## Cumulative Proportion 0.19 0.37 0.55 0.72 0.86 1.00
## 
## Mean item complexity =  1.7
## Test of the hypothesis that 6 factors are sufficient.
## 
## df null model =  153  with the objective function =  13.52 with Chi Square =  4071.79
## df of  the model are 60  and the objective function was  0.48 
## 
## The root mean square of the residuals (RMSR) is  0.02 
## The df corrected root mean square of the residuals is  0.03 
## 
## The harmonic n.obs is  309 with the empirical chi square  32.2  with prob <  1 
## The total n.obs was  309  with Likelihood Chi Square =  142.58  with prob <  1.1e-08 
## 
## Tucker Lewis Index of factoring reliability =  0.946
## RMSEA index =  0.067  and the 90 % confidence intervals are  0.053 0.081
## BIC =  -201.42
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy             
##                                                    PA5  PA4  PA3  PA1  PA2  PA6
## Correlation of (regression) scores with factors   0.97 0.95 0.93 0.93 0.93 0.96
## Multiple R square of scores with factors          0.93 0.90 0.87 0.86 0.87 0.92
## Minimum correlation of possible factor scores     0.87 0.80 0.74 0.71 0.73 0.84
fa_result8
## Factor Analysis using method =  pa
## Call: fa(r = df_1[!names(df_1) %in% c("Im15", "Im8")], nfactors = 8, 
##     rotate = "varimax", fm = "pa")
## Standardized loadings (pattern matrix) based upon correlation matrix
##       PA3  PA1  PA4  PA5  PA6  PA2  PA7  PA8   h2    u2 com
## Im1  0.23 0.18 0.19 0.07 0.90 0.08 0.13 0.18 0.99 0.010 1.5
## Im2  0.18 0.21 0.11 0.07 0.79 0.08 0.17 0.21 0.80 0.200 1.6
## Im3  0.22 0.80 0.20 0.11 0.14 0.13 0.17 0.26 0.88 0.125 1.8
## Im4  0.18 0.86 0.20 0.13 0.15 0.13 0.16 0.23 0.94 0.063 1.6
## Im5  0.18 0.63 0.19 0.06 0.21 0.17 0.18 0.07 0.58 0.425 2.1
## Im6  0.07 0.08 0.05 0.88 0.04 0.12 0.11 0.05 0.83 0.173 1.1
## Im7  0.11 0.06 0.11 0.80 0.06 0.28 0.07 0.06 0.77 0.234 1.4
## Im9  0.12 0.17 0.33 0.44 0.08 0.12 0.26 0.06 0.44 0.560 3.4
## Im10 0.05 0.16 0.20 0.22 0.05 0.91 0.05 0.15 0.98 0.020 1.4
## Im11 0.18 0.18 0.56 0.09 0.09 0.25 0.08 0.06 0.47 0.532 2.1
## Im12 0.10 0.15 0.86 0.09 0.09 0.14 0.14 0.15 0.84 0.155 1.3
## Im13 0.18 0.19 0.72 0.12 0.18 0.08 0.26 0.11 0.72 0.277 1.9
## Im14 0.06 0.16 0.20 0.28 0.11 0.78 0.05 0.14 0.79 0.211 1.6
## Im16 0.12 0.26 0.15 0.05 0.24 0.19 0.15 0.62 0.60 0.399 2.3
## Im17 0.20 0.20 0.22 0.17 0.19 0.06 0.86 0.17 0.97 0.031 1.7
## Im18 0.19 0.24 0.24 0.16 0.14 0.06 0.73 0.13 0.75 0.248 1.9
## Im19 0.16 0.28 0.20 0.14 0.25 0.16 0.18 0.69 0.76 0.242 2.4
## Im20 0.85 0.12 0.17 0.03 0.07 0.03 0.07 0.12 0.79 0.206 1.2
## Im21 0.77 0.15 0.11 0.08 0.15 0.04 0.15 0.06 0.68 0.317 1.3
## Im22 0.80 0.20 0.13 0.18 0.20 0.07 0.15 0.07 0.80 0.202 1.5
## 
##                        PA3  PA1  PA4  PA5  PA6  PA2  PA7  PA8
## SS loadings           2.38 2.35 2.17 1.94 1.83 1.79 1.70 1.22
## Proportion Var        0.12 0.12 0.11 0.10 0.09 0.09 0.08 0.06
## Cumulative Var        0.12 0.24 0.34 0.44 0.53 0.62 0.71 0.77
## Proportion Explained  0.16 0.15 0.14 0.13 0.12 0.12 0.11 0.08
## Cumulative Proportion 0.16 0.31 0.45 0.57 0.69 0.81 0.92 1.00
## 
## Mean item complexity =  1.8
## Test of the hypothesis that 8 factors are sufficient.
## 
## df null model =  190  with the objective function =  15.43 with Chi Square =  4636.93
## df of  the model are 58  and the objective function was  0.36 
## 
## The root mean square of the residuals (RMSR) is  0.01 
## The df corrected root mean square of the residuals is  0.02 
## 
## The harmonic n.obs is  309 with the empirical chi square  15.22  with prob <  1 
## The total n.obs was  309  with Likelihood Chi Square =  106.28  with prob <  0.00011 
## 
## Tucker Lewis Index of factoring reliability =  0.964
## RMSEA index =  0.052  and the 90 % confidence intervals are  0.036 0.067
## BIC =  -226.25
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy             
##                                                    PA3  PA1  PA4  PA5  PA6  PA2
## Correlation of (regression) scores with factors   0.93 0.95 0.92 0.93 0.99 0.98
## Multiple R square of scores with factors          0.87 0.91 0.85 0.87 0.98 0.96
## Minimum correlation of possible factor scores     0.75 0.82 0.70 0.73 0.96 0.92
##                                                    PA7  PA8
## Correlation of (regression) scores with factors   0.97 0.82
## Multiple R square of scores with factors          0.95 0.67
## Minimum correlation of possible factor scores     0.89 0.34

We can see that for 6 Factors Analysis, we obtain a cumulative proportion variance of 0.73. In total, the extracted factors explain 73% of the variance.

For 8 Factors Analysis, we obtain a cumulative proportion variance of 0.77. In total, the extracted factors explain 77% of the variance.

BIC is lower with 8 factors than 6 factors, therefore may allow more generalization in future sample.

We should also check the root mean square of residuals (RMSR). An acceptable value should be closer to 0. In 6 Factors Analysis we have 0.067 and in 8 Factors Analysis we have 0.052 (closer to 0).

Finally, we must check the Tucker-Lewis Index (TLI). An acceptable value must be greater over 0.9. In 6 Factors Analysis we have 0.946 and in 8 Factors Analysis we have 0.964.

Therefore 8 Factors Analysis is overall better, with better BIC, RMSR and TLI and also explain more the total variance with 77%.

Choosing the Optimal Number of Factors in Exploratory Factor Analysis: A Model Selection Perspective

Labeling 8 Factors

colnames(fa_result8$loadings) <- c("Shopping Experience", "Store Decoration","Luxury Brands","French Culture","Product Assortment","Gourmet Food","Trendiness","Professionalism")

Shopping_Experience <- c("Im20","Im21","Im22")
Store_Decoration <- c("Im3","Im4","Im5")
Luxury_Brands <- c("Im11","Im12","Im13")
French_Culture <- c("Im6","Im7","Im9")
Product_Assortment <- c("Im1","Im2")
Gourmet_Food <- c("Im10","Im14")
Trendiness <- c("Im17","Im18")
Professionalism <- c("Im16","Im19")
fa.diagram(fa_result8, sort = TRUE, adj = 1, rsize = 4, e.size = 0.061, main = "Conclusion of Factors Analysis - with 8 labeled factors", digits = 2, l.cex = 1)

Internal consistency reliability

Our next step is to assess the internal consistency reliability of the factors that were identified through the EFA. To accomplish this, we will use Cronbach’s alpha. We will evaluate the reliability of each factor individually by incorporating only the chosen items for that particular factor.

We need to look at:

1. Cronbach’s alpha

The Cronbach’s alpha indicates the internal consistency reliability. The interpretation is detailed as follows (DeVellis, 2012, pp. 95–96):

2. Corrected item-total correlation

There are four item-total correlations provided in psych. We consider these two:

r.cor = Item-total correlation, corrected for item overlap (Revelle, 2017). This is recommended by Revelle (2017).

Ideally must be > 0.5 (Hair et al., 2010)

EFA and Cronbach’s alpha

Shopping Experience

alpha.pa1 <- psych::alpha(df_1[Shopping_Experience])
alpha.pa1$total
##  raw_alpha std.alpha   G6(smc) average_r     S/N        ase     mean       sd
##  0.8947029 0.8951786 0.8514649 0.7400356 8.54004 0.01029882 4.677454 1.341113
##   median_r
##  0.7296095

raw_alpha is over 0.7 and average items correlation is above 0.5

Store Decoration

alpha.pa1 <- psych::alpha(df_1[Store_Decoration])
alpha.pa1$total
##  raw_alpha std.alpha   G6(smc) average_r      S/N         ase     mean       sd
##  0.9080535 0.9080107 0.8910999 0.7669149 9.870833 0.009512795 4.909385 1.251142
##  median_r
##  0.713708

raw_alpha is over 0.7 and average items correlation is above 0.5

Luxury Brands

alpha.pa1 <- psych::alpha(df_1[Luxury_Brands])
alpha.pa1$total
##  raw_alpha std.alpha   G6(smc) average_r      S/N        ase     mean       sd
##  0.8362383 0.8369592 0.7932492  0.631152 5.133432 0.01631641 5.549083 1.033799
##   median_r
##  0.5951255

raw_alpha is over 0.7 and average items correlation is above 0.5

French Culture

alpha.pa1 <- psych::alpha(df_1[French_Culture])
alpha.pa1$total
##  raw_alpha std.alpha   G6(smc) average_r      S/N        ase     mean       sd
##  0.7974734 0.8043581 0.7661015 0.5781409 4.111379 0.02082013 5.532902 1.061296
##   median_r
##  0.4812433

raw_alpha is over 0.7 and average items correlation is above 0.5

Product Assortment

alpha.pa1 <- psych::alpha(df_1[Product_Assortment])
alpha.pa1$total
##  raw_alpha std.alpha   G6(smc) average_r      S/N         ase     mean      sd
##  0.9370045 0.9377564 0.8828073 0.8828073 15.06591 0.007118763 4.847896 1.27965
##   median_r
##  0.8828073

raw_alpha is over 0.7 and average items correlation is above 0.5

Gourmet Food

alpha.pa1 <- psych::alpha(df_1[Gourmet_Food])
alpha.pa1$total
##  raw_alpha std.alpha   G6(smc) average_r     S/N         ase     mean        sd
##  0.9327078 0.9327084 0.8739021 0.8739021 13.8607 0.007656198 6.106796 0.8498963
##   median_r
##  0.8739021

raw_alpha is over 0.7 and average items correlation is above 0.5

Trendiness

alpha.pa1 <- psych::alpha(df_1[Trendiness])
alpha.pa1$total
##  raw_alpha std.alpha   G6(smc) average_r      S/N         ase     mean       sd
##  0.9155341 0.9175086 0.8475898 0.8475898 11.12248 0.009476765 4.737864 1.287763
##   median_r
##  0.8475898

raw_alpha is over 0.7 and average items correlation is above 0.5

Professionalism

alpha.pa1 <- psych::alpha(df_1[Professionalism])
alpha.pa1$total
##  raw_alpha std.alpha   G6(smc) average_r      S/N        ase     mean       sd
##  0.8027054 0.8029503 0.6707744 0.6707744 4.074861 0.02242662 5.082524 1.119696
##   median_r
##  0.6707744

raw_alpha is over 0.7 and average items correlation is above 0.5

Our assessment suggests that the factors extracted are reliable, and therefore it is advisable to retain all the items related to these factors.

Dimensions by which Galeries Layfayette is perceived?

fa.diagram(fa_result8, sort = TRUE, adj = 1, rsize = 4, e.size = 0.061, main = "Galeries Lafayette - Perception Dimensions", digits = 2, l.cex = 1)

Dimensions Definitions:

Product Assortment: This group pertains to the variety and range of products offered by the store.

Store Decoration: This group pertains to the aesthetic elements of the store’s interior and exterior, such as the artistic and creative decoration of the sales area, and the appealing arrangement of shop windows.

French Culture: This group pertains to elements of French culture, such as French savoir-vivre, fashion.

Gourmet Food: This group pertains to high-quality food and cosmetic products offered by the store.

Luxury Brands: This group pertains to the presence of luxury and designer brands in the store.

Professionalism: This group pertains to elements of professionalism, such as the store’s professional appearance towards customers and professional organization.

Trendiness: This group pertains to the store’s ability to stay current and up-to-date with the latest trends in the market.

Shopping Experience: This group pertains to the overall shopping experience, including elements such as relaxing shopping, a great place to stroll, and an intimate shop atmosphere.

Confirmatory Factor Analysis

From Assistant For confirmatory factor analysis (CFA) and structural equation modeling (SEM), please use the raw data (which includes the missing values) to perform CFA and SEM, and use maximum likelihood (ML) to handle the missing data.

df <- read.csv2('Case Study III_Structural Equation Modeling.csv', na.strings = '999', sep = ',')
model_CFA <-"
Shopping_Experience =~ Im20+Im21+Im22
Store_Decoration =~ Im3+Im4+Im5
Luxury_Brands =~ Im11+Im12+Im13
French_Culture =~ Im6+Im7+Im9
Product_Assortment =~ Im1+Im2
Gourmet_Food =~ Im10+Im14
Trendiness =~ Im17+Im18
Professionalism =~ Im16+Im19"

fit_CFA <- lavaan::cfa(model_CFA, data=df, missing="ML")

summary(fit_CFA,fit.measures=TRUE, standardized=TRUE)
## lavaan 0.6.15 ended normally after 110 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        88
## 
##   Number of observations                           553
##   Number of missing patterns                        82
## 
## Model Test User Model:
##                                                       
##   Test statistic                               383.534
##   Degrees of freedom                               142
##   P-value (Chi-square)                           0.000
## 
## Model Test Baseline Model:
## 
##   Test statistic                              7789.413
##   Degrees of freedom                               190
##   P-value                                        0.000
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.968
##   Tucker-Lewis Index (TLI)                       0.957
##                                                       
##   Robust Comparative Fit Index (CFI)             0.968
##   Robust Tucker-Lewis Index (TLI)                0.957
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)             -13802.030
##   Loglikelihood unrestricted model (H1)     -13610.263
##                                                       
##   Akaike (AIC)                               27780.060
##   Bayesian (BIC)                             28159.811
##   Sample-size adjusted Bayesian (SABIC)      27880.460
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.055
##   90 Percent confidence interval - lower         0.049
##   90 Percent confidence interval - upper         0.062
##   P-value H_0: RMSEA <= 0.050                    0.087
##   P-value H_0: RMSEA >= 0.080                    0.000
##                                                       
##   Robust RMSEA                                   0.057
##   90 Percent confidence interval - lower         0.050
##   90 Percent confidence interval - upper         0.064
##   P-value H_0: Robust RMSEA <= 0.050             0.055
##   P-value H_0: Robust RMSEA >= 0.080             0.000
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.052
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Observed
##   Observed information based on                Hessian
## 
## Latent Variables:
##                          Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   Shopping_Experience =~                                                      
##     Im20                    1.000                               1.264    0.845
##     Im21                    0.849    0.041   20.818    0.000    1.073    0.783
##     Im22                    1.060    0.047   22.590    0.000    1.341    0.877
##   Store_Decoration =~                                                         
##     Im3                     1.000                               1.236    0.937
##     Im4                     1.056    0.025   42.717    0.000    1.305    0.969
##     Im5                     0.818    0.034   23.813    0.000    1.011    0.760
##   Luxury_Brands =~                                                            
##     Im11                    1.000                               0.703    0.615
##     Im12                    1.410    0.094   15.048    0.000    0.991    0.872
##     Im13                    1.464    0.105   13.971    0.000    1.029    0.855
##   French_Culture =~                                                           
##     Im6                     1.000                               1.002    0.835
##     Im7                     1.107    0.050   22.219    0.000    1.109    0.919
##     Im9                     0.789    0.057   13.916    0.000    0.790    0.585
##   Product_Assortment =~                                                       
##     Im1                     1.000                               1.305    0.980
##     Im2                     0.885    0.033   27.039    0.000    1.155    0.899
##   Gourmet_Food =~                                                             
##     Im10                    1.000                               0.812    0.924
##     Im14                    1.014    0.035   28.587    0.000    0.823    0.952
##   Trendiness =~                                                               
##     Im17                    1.000                               1.204    0.968
##     Im18                    0.995    0.041   24.250    0.000    1.197    0.857
##   Professionalism =~                                                          
##     Im16                    1.000                               0.922    0.766
##     Im19                    1.045    0.061   17.188    0.000    0.963    0.856
## 
## Covariances:
##                          Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   Shopping_Experience ~~                                                      
##     Store_Decoratn          0.729    0.082    8.911    0.000    0.467    0.467
##     Luxury_Brands           0.372    0.053    7.009    0.000    0.418    0.418
##     French_Culture          0.446    0.066    6.749    0.000    0.352    0.352
##     Prdct_Assrtmnt          0.739    0.085    8.728    0.000    0.448    0.448
##     Gourmet_Food            0.303    0.051    5.951    0.000    0.295    0.295
##     Trendiness              0.786    0.081    9.715    0.000    0.517    0.517
##     Professionalsm          0.557    0.069    8.091    0.000    0.478    0.478
##   Store_Decoration ~~                                                         
##     Luxury_Brands           0.409    0.051    8.040    0.000    0.471    0.471
##     French_Culture          0.449    0.063    7.099    0.000    0.363    0.363
##     Prdct_Assrtmnt          0.711    0.079    9.032    0.000    0.440    0.440
##     Gourmet_Food            0.418    0.050    8.401    0.000    0.417    0.417
##     Trendiness              0.770    0.076   10.141    0.000    0.517    0.517
##     Professionalsm          0.744    0.071   10.469    0.000    0.653    0.653
##   Luxury_Brands ~~                                                            
##     French_Culture          0.239    0.039    6.077    0.000    0.340    0.340
##     Prdct_Assrtmnt          0.438    0.054    8.161    0.000    0.478    0.478
##     Gourmet_Food            0.258    0.034    7.665    0.000    0.452    0.452
##     Trendiness              0.479    0.053    9.044    0.000    0.566    0.566
##     Professionalsm          0.343    0.043    7.947    0.000    0.529    0.529
##   French_Culture ~~                                                           
##     Prdct_Assrtmnt          0.321    0.063    5.121    0.000    0.246    0.246
##     Gourmet_Food            0.490    0.047   10.536    0.000    0.603    0.603
##     Trendiness              0.439    0.062    7.087    0.000    0.364    0.364
##     Professionalsm          0.360    0.052    6.937    0.000    0.391    0.391
##   Product_Assortment ~~                                                       
##     Gourmet_Food            0.328    0.050    6.581    0.000    0.309    0.309
##     Trendiness              0.817    0.079   10.362    0.000    0.519    0.519
##     Professionalsm          0.718    0.072    9.961    0.000    0.597    0.597
##   Gourmet_Food ~~                                                             
##     Trendiness              0.318    0.047    6.804    0.000    0.325    0.325
##     Professionalsm          0.373    0.043    8.600    0.000    0.498    0.498
##   Trendiness ~~                                                               
##     Professionalsm          0.667    0.066   10.043    0.000    0.601    0.601
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .Im20              4.672    0.064   73.178    0.000    4.672    3.123
##    .Im21              5.139    0.058   87.970    0.000    5.139    3.751
##    .Im22              4.279    0.065   65.401    0.000    4.279    2.799
##    .Im3               4.995    0.056   88.565    0.000    4.995    3.786
##    .Im4               4.999    0.057   86.988    0.000    4.999    3.712
##    .Im5               5.035    0.057   87.848    0.000    5.035    3.787
##    .Im11              5.653    0.049  115.273    0.000    5.653    4.943
##    .Im12              5.666    0.049  116.092    0.000    5.666    4.983
##    .Im13              5.448    0.052  105.619    0.000    5.448    4.524
##    .Im6               5.826    0.051  113.774    0.000    5.826    4.856
##    .Im7               5.751    0.052  111.068    0.000    5.751    4.766
##    .Im9               5.075    0.058   87.408    0.000    5.075    3.756
##    .Im1               4.790    0.057   84.203    0.000    4.790    3.597
##    .Im2               4.857    0.055   88.356    0.000    4.857    3.779
##    .Im10              6.100    0.037  162.799    0.000    6.100    6.937
##    .Im14              6.138    0.037  165.865    0.000    6.138    7.093
##    .Im17              5.025    0.053   94.529    0.000    5.025    4.041
##    .Im18              4.595    0.060   76.455    0.000    4.595    3.287
##    .Im16              5.135    0.052   99.150    0.000    5.135    4.269
##    .Im19              5.145    0.048  106.953    0.000    5.145    4.574
##     Shoppng_Exprnc    0.000                               0.000    0.000
##     Store_Decoratn    0.000                               0.000    0.000
##     Luxury_Brands     0.000                               0.000    0.000
##     French_Culture    0.000                               0.000    0.000
##     Prdct_Assrtmnt    0.000                               0.000    0.000
##     Gourmet_Food      0.000                               0.000    0.000
##     Trendiness        0.000                               0.000    0.000
##     Professionalsm    0.000                               0.000    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .Im20              0.639    0.061   10.445    0.000    0.639    0.285
##    .Im21              0.726    0.057   12.672    0.000    0.726    0.387
##    .Im22              0.540    0.063    8.515    0.000    0.540    0.231
##    .Im3               0.213    0.024    8.760    0.000    0.213    0.123
##    .Im4               0.109    0.024    4.524    0.000    0.109    0.060
##    .Im5               0.747    0.049   15.217    0.000    0.747    0.422
##    .Im11              0.814    0.055   14.803    0.000    0.814    0.622
##    .Im12              0.310    0.040    7.833    0.000    0.310    0.239
##    .Im13              0.390    0.045    8.771    0.000    0.390    0.269
##    .Im6               0.436    0.042   10.443    0.000    0.436    0.303
##    .Im7               0.227    0.042    5.343    0.000    0.227    0.156
##    .Im9               1.201    0.080   15.032    0.000    1.201    0.658
##    .Im1               0.070    0.050    1.383    0.167    0.070    0.039
##    .Im2               0.317    0.044    7.242    0.000    0.317    0.192
##    .Im10              0.113    0.019    5.935    0.000    0.113    0.146
##    .Im14              0.071    0.019    3.764    0.000    0.071    0.094
##    .Im17              0.096    0.045    2.153    0.031    0.096    0.062
##    .Im18              0.520    0.054    9.566    0.000    0.520    0.266
##    .Im16              0.598    0.052   11.498    0.000    0.598    0.413
##    .Im19              0.338    0.045    7.481    0.000    0.338    0.267
##     Shoppng_Exprnc    1.599    0.138   11.620    0.000    1.000    1.000
##     Store_Decoratn    1.527    0.107   14.325    0.000    1.000    1.000
##     Luxury_Brands     0.494    0.067    7.363    0.000    1.000    1.000
##     French_Culture    1.003    0.089   11.297    0.000    1.000    1.000
##     Prdct_Assrtmnt    1.704    0.118   14.391    0.000    1.000    1.000
##     Gourmet_Food      0.660    0.049   13.357    0.000    1.000    1.000
##     Trendiness        1.450    0.104   13.998    0.000    1.000    1.000
##     Professionalsm    0.849    0.088    9.644    0.000    1.000    1.000
standardizedsolution(fit_CFA)
##                     lhs op                 rhs est.std    se       z pvalue
## 1   Shopping_Experience =~                Im20   0.845 0.017  48.438  0.000
## 2   Shopping_Experience =~                Im21   0.783 0.021  37.806  0.000
## 3   Shopping_Experience =~                Im22   0.877 0.017  53.040  0.000
## 4      Store_Decoration =~                 Im3   0.937 0.008 114.004  0.000
## 5      Store_Decoration =~                 Im4   0.969 0.007 137.088  0.000
## 6      Store_Decoration =~                 Im5   0.760 0.019  39.237  0.000
## 7         Luxury_Brands =~                Im11   0.615 0.031  19.967  0.000
## 8         Luxury_Brands =~                Im12   0.872 0.019  47.007  0.000
## 9         Luxury_Brands =~                Im13   0.855 0.019  44.643  0.000
## 10       French_Culture =~                 Im6   0.835 0.019  44.725  0.000
## 11       French_Culture =~                 Im7   0.919 0.016  55.930  0.000
## 12       French_Culture =~                 Im9   0.585 0.032  18.290  0.000
## 13   Product_Assortment =~                 Im1   0.980 0.015  67.452  0.000
## 14   Product_Assortment =~                 Im2   0.899 0.016  57.629  0.000
## 15         Gourmet_Food =~                Im10   0.924 0.014  66.312  0.000
## 16         Gourmet_Food =~                Im14   0.952 0.013  70.640  0.000
## 17           Trendiness =~                Im17   0.968 0.015  64.310  0.000
## 18           Trendiness =~                Im18   0.857 0.017  49.115  0.000
## 19      Professionalism =~                Im16   0.766 0.024  31.511  0.000
## 20      Professionalism =~                Im19   0.856 0.022  39.167  0.000
## 21                 Im20 ~~                Im20   0.285 0.030   9.674  0.000
## 22                 Im21 ~~                Im21   0.387 0.032  11.912  0.000
## 23                 Im22 ~~                Im22   0.231 0.029   7.969  0.000
## 24                  Im3 ~~                 Im3   0.123 0.015   7.962  0.000
## 25                  Im4 ~~                 Im4   0.060 0.014   4.381  0.000
## 26                  Im5 ~~                 Im5   0.422 0.029  14.339  0.000
## 27                 Im11 ~~                Im11   0.622 0.038  16.446  0.000
## 28                 Im12 ~~                Im12   0.239 0.032   7.401  0.000
## 29                 Im13 ~~                Im13   0.269 0.033   8.226  0.000
## 30                  Im6 ~~                 Im6   0.303 0.031   9.724  0.000
## 31                  Im7 ~~                 Im7   0.156 0.030   5.154  0.000
## 32                  Im9 ~~                 Im9   0.658 0.037  17.595  0.000
## 33                  Im1 ~~                 Im1   0.039 0.028   1.379  0.168
## 34                  Im2 ~~                 Im2   0.192 0.028   6.855  0.000
## 35                 Im10 ~~                Im10   0.146 0.026   5.685  0.000
## 36                 Im14 ~~                Im14   0.094 0.026   3.684  0.000
## 37                 Im17 ~~                Im17   0.062 0.029   2.137  0.033
## 38                 Im18 ~~                Im18   0.266 0.030   8.905  0.000
## 39                 Im16 ~~                Im16   0.413 0.037  11.093  0.000
## 40                 Im19 ~~                Im19   0.267 0.037   7.143  0.000
## 41  Shopping_Experience ~~ Shopping_Experience   1.000 0.000      NA     NA
## 42     Store_Decoration ~~    Store_Decoration   1.000 0.000      NA     NA
## 43        Luxury_Brands ~~       Luxury_Brands   1.000 0.000      NA     NA
## 44       French_Culture ~~      French_Culture   1.000 0.000      NA     NA
## 45   Product_Assortment ~~  Product_Assortment   1.000 0.000      NA     NA
## 46         Gourmet_Food ~~        Gourmet_Food   1.000 0.000      NA     NA
## 47           Trendiness ~~          Trendiness   1.000 0.000      NA     NA
## 48      Professionalism ~~     Professionalism   1.000 0.000      NA     NA
## 49  Shopping_Experience ~~    Store_Decoration   0.467 0.038  12.442  0.000
## 50  Shopping_Experience ~~       Luxury_Brands   0.418 0.042   9.856  0.000
## 51  Shopping_Experience ~~      French_Culture   0.352 0.043   8.118  0.000
## 52  Shopping_Experience ~~  Product_Assortment   0.448 0.038  11.709  0.000
## 53  Shopping_Experience ~~        Gourmet_Food   0.295 0.043   6.792  0.000
## 54  Shopping_Experience ~~          Trendiness   0.517 0.037  14.113  0.000
## 55  Shopping_Experience ~~     Professionalism   0.478 0.042  11.468  0.000
## 56     Store_Decoration ~~       Luxury_Brands   0.471 0.038  12.457  0.000
## 57     Store_Decoration ~~      French_Culture   0.363 0.041   8.789  0.000
## 58     Store_Decoration ~~  Product_Assortment   0.440 0.036  12.074  0.000
## 59     Store_Decoration ~~        Gourmet_Food   0.417 0.038  11.052  0.000
## 60     Store_Decoration ~~          Trendiness   0.517 0.035  14.863  0.000
## 61     Store_Decoration ~~     Professionalism   0.653 0.032  20.635  0.000
## 62        Luxury_Brands ~~      French_Culture   0.340 0.044   7.657  0.000
## 63        Luxury_Brands ~~  Product_Assortment   0.478 0.038  12.618  0.000
## 64        Luxury_Brands ~~        Gourmet_Food   0.452 0.039  11.626  0.000
## 65        Luxury_Brands ~~          Trendiness   0.566 0.035  16.258  0.000
## 66        Luxury_Brands ~~     Professionalism   0.529 0.040  13.232  0.000
## 67       French_Culture ~~  Product_Assortment   0.246 0.044   5.579  0.000
## 68       French_Culture ~~        Gourmet_Food   0.603 0.032  18.938  0.000
## 69       French_Culture ~~          Trendiness   0.364 0.042   8.656  0.000
## 70       French_Culture ~~     Professionalism   0.391 0.045   8.769  0.000
## 71   Product_Assortment ~~        Gourmet_Food   0.309 0.041   7.490  0.000
## 72   Product_Assortment ~~          Trendiness   0.519 0.034  15.157  0.000
## 73   Product_Assortment ~~     Professionalism   0.597 0.036  16.775  0.000
## 74         Gourmet_Food ~~          Trendiness   0.325 0.041   7.939  0.000
## 75         Gourmet_Food ~~     Professionalism   0.498 0.039  12.700  0.000
## 76           Trendiness ~~     Professionalism   0.601 0.035  17.348  0.000
## 77                 Im20 ~1                       3.123 0.103  30.213  0.000
## 78                 Im21 ~1                       3.751 0.121  31.049  0.000
## 79                 Im22 ~1                       2.799 0.095  29.400  0.000
## 80                  Im3 ~1                       3.786 0.122  30.965  0.000
## 81                  Im4 ~1                       3.712 0.120  30.889  0.000
## 82                  Im5 ~1                       3.787 0.124  30.535  0.000
## 83                 Im11 ~1                       4.943 0.157  31.542  0.000
## 84                 Im12 ~1                       4.983 0.158  31.525  0.000
## 85                 Im13 ~1                       4.524 0.144  31.341  0.000
## 86                  Im6 ~1                       4.856 0.153  31.831  0.000
## 87                  Im7 ~1                       4.766 0.151  31.497  0.000
## 88                  Im9 ~1                       3.756 0.122  30.740  0.000
## 89                  Im1 ~1                       3.597 0.117  30.765  0.000
## 90                  Im2 ~1                       3.779 0.122  30.960  0.000
## 91                 Im10 ~1                       6.937 0.213  32.501  0.000
## 92                 Im14 ~1                       7.093 0.221  32.063  0.000
## 93                 Im17 ~1                       4.041 0.130  31.175  0.000
## 94                 Im18 ~1                       3.287 0.109  30.205  0.000
## 95                 Im16 ~1                       4.269 0.137  31.187  0.000
## 96                 Im19 ~1                       4.574 0.145  31.501  0.000
## 97  Shopping_Experience ~1                       0.000 0.000      NA     NA
## 98     Store_Decoration ~1                       0.000 0.000      NA     NA
## 99        Luxury_Brands ~1                       0.000 0.000      NA     NA
## 100      French_Culture ~1                       0.000 0.000      NA     NA
## 101  Product_Assortment ~1                       0.000 0.000      NA     NA
## 102        Gourmet_Food ~1                       0.000 0.000      NA     NA
## 103          Trendiness ~1                       0.000 0.000      NA     NA
## 104     Professionalism ~1                       0.000 0.000      NA     NA
##     ci.lower ci.upper
## 1      0.811    0.880
## 2      0.743    0.824
## 3      0.844    0.909
## 4      0.921    0.953
## 5      0.956    0.983
## 6      0.722    0.798
## 7      0.554    0.675
## 8      0.836    0.908
## 9      0.817    0.892
## 10     0.798    0.871
## 11     0.887    0.951
## 12     0.522    0.647
## 13     0.952    1.009
## 14     0.868    0.929
## 15     0.897    0.951
## 16     0.925    0.978
## 17     0.939    0.998
## 18     0.822    0.891
## 19     0.718    0.814
## 20     0.813    0.899
## 21     0.228    0.343
## 22     0.323    0.450
## 23     0.174    0.288
## 24     0.092    0.153
## 25     0.033    0.087
## 26     0.365    0.480
## 27     0.548    0.696
## 28     0.176    0.303
## 29     0.205    0.333
## 30     0.242    0.364
## 31     0.096    0.215
## 32     0.585    0.731
## 33    -0.017    0.095
## 34     0.137    0.247
## 35     0.096    0.197
## 36     0.044    0.145
## 37     0.005    0.119
## 38     0.208    0.325
## 39     0.340    0.486
## 40     0.194    0.341
## 41     1.000    1.000
## 42     1.000    1.000
## 43     1.000    1.000
## 44     1.000    1.000
## 45     1.000    1.000
## 46     1.000    1.000
## 47     1.000    1.000
## 48     1.000    1.000
## 49     0.393    0.540
## 50     0.335    0.501
## 51     0.267    0.437
## 52     0.373    0.523
## 53     0.210    0.380
## 54     0.445    0.588
## 55     0.396    0.560
## 56     0.397    0.545
## 57     0.282    0.444
## 58     0.369    0.512
## 59     0.343    0.490
## 60     0.449    0.586
## 61     0.591    0.715
## 62     0.253    0.426
## 63     0.404    0.552
## 64     0.376    0.529
## 65     0.498    0.635
## 66     0.451    0.607
## 67     0.159    0.332
## 68     0.540    0.665
## 69     0.282    0.447
## 70     0.303    0.478
## 71     0.228    0.390
## 72     0.452    0.587
## 73     0.527    0.666
## 74     0.245    0.406
## 75     0.421    0.575
## 76     0.533    0.669
## 77     2.921    3.326
## 78     3.514    3.987
## 79     2.612    2.985
## 80     3.546    4.026
## 81     3.477    3.948
## 82     3.544    4.030
## 83     4.636    5.250
## 84     4.673    5.293
## 85     4.241    4.807
## 86     4.557    5.155
## 87     4.469    5.062
## 88     3.517    3.995
## 89     3.368    3.826
## 90     3.540    4.018
## 91     6.519    7.355
## 92     6.660    7.527
## 93     3.787    4.295
## 94     3.074    3.500
## 95     4.001    4.537
## 96     4.289    4.859
## 97     0.000    0.000
## 98     0.000    0.000
## 99     0.000    0.000
## 100    0.000    0.000
## 101    0.000    0.000
## 102    0.000    0.000
## 103    0.000    0.000
## 104    0.000    0.000

CFA Visualization

lavaanPlot(model = fit_CFA, node_options = list(shape = "box", fontname = "Helvetica"), edge_options = list(color = "palegreen4"), coefs = TRUE, sig = 0.05, covs = FALSE, digits = 2)
lavaanPlot(model = fit_CFA, node_options = list(shape = "box", fontname = "Helvetica"), edge_options = list(color = "palegreen4"), coefs = TRUE,sig = 0.05, covs = TRUE, digits = 2)
#modificationindices(fit) %>% filter(mi>10)

Structure Equation Modelling

model_SEM <- " 
Shopping_Experience =~ Im20+Im21+Im22
Store_Decoration =~ Im3+Im4+Im5
Luxury_Brands =~ Im11+Im12+Im13
French_Culture =~ Im6+Im7+Im9
Product_Assortment =~ Im1+Im2
Gourmet_Food =~ Im10+Im14
Trendiness =~ Im17+Im18
Professionalism =~ Im16+Im19

CS =~ SAT_1 + SAT_2 + SAT_3
AC =~ COM_A1 + COM_A2 + COM_A3 + COM_A4

RI =~ C_REP1 + C_REP2 + C_REP3
CI =~ C_CR1 + C_CR3 + C_CR4

CS ~ Shopping_Experience + Store_Decoration + Luxury_Brands + French_Culture + Product_Assortment + Gourmet_Food + Trendiness + Professionalism
AC ~ Shopping_Experience + Store_Decoration + Luxury_Brands + French_Culture + Product_Assortment + Gourmet_Food + Trendiness + Professionalism

RI ~ CS + AC + Shopping_Experience + Store_Decoration + Luxury_Brands + French_Culture + Product_Assortment + Gourmet_Food + Trendiness + Professionalism
CI ~ CS + AC + Shopping_Experience + Store_Decoration + Luxury_Brands + French_Culture + Product_Assortment + Gourmet_Food + Trendiness + Professionalism "


fit_SEM <- lavaan::cfa(model_SEM, data=df, missing="ML")

summary(fit_SEM,fit.measures=TRUE, standardized=TRUE)
## lavaan 0.6.15 ended normally after 153 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                       164
## 
##   Number of observations                           553
##   Number of missing patterns                       137
## 
## Model Test User Model:
##                                                       
##   Test statistic                               835.621
##   Degrees of freedom                               430
##   P-value (Chi-square)                           0.000
## 
## Model Test Baseline Model:
## 
##   Test statistic                             12305.018
##   Degrees of freedom                               528
##   P-value                                        0.000
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.966
##   Tucker-Lewis Index (TLI)                       0.958
##                                                       
##   Robust Comparative Fit Index (CFI)             0.966
##   Robust Tucker-Lewis Index (TLI)                0.958
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)             -23197.252
##   Loglikelihood unrestricted model (H1)     -22779.442
##                                                       
##   Akaike (AIC)                               46722.504
##   Bayesian (BIC)                             47430.223
##   Sample-size adjusted Bayesian (SABIC)      46909.614
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.041
##   90 Percent confidence interval - lower         0.037
##   90 Percent confidence interval - upper         0.045
##   P-value H_0: RMSEA <= 0.050                    1.000
##   P-value H_0: RMSEA >= 0.080                    0.000
##                                                       
##   Robust RMSEA                                   0.042
##   90 Percent confidence interval - lower         0.038
##   90 Percent confidence interval - upper         0.046
##   P-value H_0: Robust RMSEA <= 0.050             0.999
##   P-value H_0: Robust RMSEA >= 0.080             0.000
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.048
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Observed
##   Observed information based on                Hessian
## 
## Latent Variables:
##                          Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   Shopping_Experience =~                                                      
##     Im20                    1.000                               1.262    0.844
##     Im21                    0.857    0.041   20.998    0.000    1.081    0.789
##     Im22                    1.056    0.046   23.027    0.000    1.333    0.873
##   Store_Decoration =~                                                         
##     Im3                     1.000                               1.235    0.936
##     Im4                     1.057    0.025   42.733    0.000    1.306    0.970
##     Im5                     0.818    0.034   23.805    0.000    1.010    0.760
##   Luxury_Brands =~                                                            
##     Im11                    1.000                               0.700    0.613
##     Im12                    1.415    0.094   15.007    0.000    0.991    0.872
##     Im13                    1.468    0.105   13.929    0.000    1.029    0.855
##   French_Culture =~                                                           
##     Im6                     1.000                               1.004    0.837
##     Im7                     1.101    0.049   22.598    0.000    1.106    0.916
##     Im9                     0.788    0.057   13.939    0.000    0.792    0.586
##   Product_Assortment =~                                                       
##     Im1                     1.000                               1.297    0.974
##     Im2                     0.895    0.032   28.321    0.000    1.161    0.904
##   Gourmet_Food =~                                                             
##     Im10                    1.000                               0.811    0.922
##     Im14                    1.018    0.035   28.748    0.000    0.826    0.954
##   Trendiness =~                                                               
##     Im17                    1.000                               1.205    0.969
##     Im18                    0.993    0.041   24.204    0.000    1.196    0.856
##   Professionalism =~                                                          
##     Im16                    1.000                               0.919    0.764
##     Im19                    1.043    0.058   17.879    0.000    0.959    0.853
##   CS =~                                                                       
##     SAT_1                   1.000                               0.882    0.865
##     SAT_2                   0.933    0.049   18.916    0.000    0.823    0.819
##     SAT_3                   0.809    0.055   14.802    0.000    0.714    0.624
##   AC =~                                                                       
##     COM_A1                  1.000                               1.144    0.796
##     COM_A2                  1.174    0.055   21.504    0.000    1.343    0.836
##     COM_A3                  1.162    0.058   20.027    0.000    1.329    0.817
##     COM_A4                  1.278    0.061   20.800    0.000    1.462    0.842
##   RI =~                                                                       
##     C_REP1                  1.000                               0.596    0.816
##     C_REP2                  0.971    0.043   22.489    0.000    0.579    0.931
##     C_REP3                  0.702    0.037   19.036    0.000    0.419    0.756
##   CI =~                                                                       
##     C_CR1                   1.000                               1.658    0.851
##     C_CR3                   1.033    0.051   20.247    0.000    1.712    0.826
##     C_CR4                   0.964    0.049   19.766    0.000    1.598    0.806
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   CS ~                                                                  
##     Shoppng_Exprnc    0.051    0.038    1.357    0.175    0.074    0.074
##     Store_Decoratn   -0.110    0.043   -2.551    0.011   -0.153   -0.153
##     Luxury_Brands    -0.041    0.075   -0.543    0.587   -0.032   -0.032
##     French_Culture    0.109    0.050    2.156    0.031    0.124    0.124
##     Prdct_Assrtmnt    0.135    0.040    3.403    0.001    0.198    0.198
##     Gourmet_Food      0.075    0.066    1.147    0.251    0.069    0.069
##     Trendiness        0.004    0.045    0.089    0.929    0.005    0.005
##     Professionalsm    0.461    0.088    5.259    0.000    0.480    0.480
##   AC ~                                                                  
##     Shoppng_Exprnc    0.372    0.052    7.186    0.000    0.410    0.410
##     Store_Decoratn   -0.026    0.054   -0.480    0.631   -0.028   -0.028
##     Luxury_Brands    -0.193    0.098   -1.959    0.050   -0.118   -0.118
##     French_Culture    0.237    0.065    3.621    0.000    0.208    0.208
##     Prdct_Assrtmnt    0.102    0.050    2.041    0.041    0.116    0.116
##     Gourmet_Food      0.016    0.085    0.194    0.846    0.012    0.012
##     Trendiness       -0.026    0.058   -0.450    0.653   -0.028   -0.028
##     Professionalsm    0.162    0.105    1.541    0.123    0.131    0.131
##   RI ~                                                                  
##     CS                0.215    0.045    4.785    0.000    0.318    0.318
##     AC                0.186    0.030    6.164    0.000    0.356    0.356
##     Shoppng_Exprnc    0.040    0.028    1.435    0.151    0.086    0.086
##     Store_Decoratn    0.010    0.029    0.353    0.724    0.021    0.021
##     Luxury_Brands     0.078    0.051    1.523    0.128    0.092    0.092
##     French_Culture   -0.041    0.034   -1.187    0.235   -0.069   -0.069
##     Prdct_Assrtmnt   -0.017    0.026   -0.675    0.500   -0.038   -0.038
##     Gourmet_Food      0.042    0.044    0.963    0.335    0.057    0.057
##     Trendiness       -0.009    0.030   -0.295    0.768   -0.018   -0.018
##     Professionalsm   -0.037    0.060   -0.621    0.535   -0.058   -0.058
##   CI ~                                                                  
##     CS               -0.356    0.131   -2.710    0.007   -0.190   -0.190
##     AC                0.548    0.091    6.021    0.000    0.378    0.378
##     Shoppng_Exprnc    0.152    0.087    1.738    0.082    0.116    0.116
##     Store_Decoratn   -0.030    0.090   -0.331    0.741   -0.022   -0.022
##     Luxury_Brands     0.201    0.159    1.264    0.206    0.085    0.085
##     French_Culture   -0.134    0.107   -1.254    0.210   -0.081   -0.081
##     Prdct_Assrtmnt   -0.007    0.080   -0.091    0.927   -0.006   -0.006
##     Gourmet_Food     -0.074    0.136   -0.542    0.588   -0.036   -0.036
##     Trendiness        0.026    0.093    0.286    0.775    0.019    0.019
##     Professionalsm   -0.178    0.184   -0.967    0.334   -0.099   -0.099
## 
## Covariances:
##                          Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   Shopping_Experience ~~                                                      
##     Store_Decoratn          0.728    0.082    8.916    0.000    0.467    0.467
##     Luxury_Brands           0.370    0.053    7.012    0.000    0.418    0.418
##     French_Culture          0.446    0.066    6.743    0.000    0.352    0.352
##     Prdct_Assrtmnt          0.732    0.084    8.676    0.000    0.447    0.447
##     Gourmet_Food            0.302    0.051    5.952    0.000    0.295    0.295
##     Trendiness              0.784    0.081    9.709    0.000    0.516    0.516
##     Professionalsm          0.552    0.068    8.107    0.000    0.476    0.476
##   Store_Decoration ~~                                                         
##     Luxury_Brands           0.407    0.051    8.023    0.000    0.470    0.470
##     French_Culture          0.452    0.063    7.146    0.000    0.364    0.364
##     Prdct_Assrtmnt          0.708    0.079    9.017    0.000    0.442    0.442
##     Gourmet_Food            0.417    0.050    8.396    0.000    0.417    0.417
##     Trendiness              0.769    0.076   10.130    0.000    0.516    0.516
##     Professionalsm          0.744    0.070   10.552    0.000    0.655    0.655
##   Luxury_Brands ~~                                                            
##     French_Culture          0.239    0.039    6.077    0.000    0.339    0.339
##     Prdct_Assrtmnt          0.433    0.053    8.112    0.000    0.477    0.477
##     Gourmet_Food            0.257    0.034    7.648    0.000    0.452    0.452
##     Trendiness              0.477    0.053    9.027    0.000    0.565    0.565
##     Professionalsm          0.342    0.043    7.967    0.000    0.531    0.531
##   French_Culture ~~                                                           
##     Prdct_Assrtmnt          0.323    0.063    5.160    0.000    0.248    0.248
##     Gourmet_Food            0.490    0.047   10.536    0.000    0.602    0.602
##     Trendiness              0.443    0.062    7.144    0.000    0.366    0.366
##     Professionalsm          0.362    0.052    6.979    0.000    0.392    0.392
##   Product_Assortment ~~                                                       
##     Gourmet_Food            0.328    0.050    6.606    0.000    0.312    0.312
##     Trendiness              0.814    0.079   10.355    0.000    0.521    0.521
##     Professionalsm          0.717    0.071   10.051    0.000    0.602    0.602
##   Gourmet_Food ~~                                                             
##     Trendiness              0.317    0.047    6.800    0.000    0.325    0.325
##     Professionalsm          0.372    0.043    8.640    0.000    0.500    0.500
##   Trendiness ~~                                                               
##     Professionalsm          0.667    0.066   10.108    0.000    0.602    0.602
##  .RI ~~                                                                       
##    .CI                     -0.015    0.038   -0.404    0.686   -0.021   -0.021
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .Im20              4.672    0.064   73.218    0.000    4.672    3.125
##    .Im21              5.139    0.058   87.977    0.000    5.139    3.750
##    .Im22              4.280    0.065   65.479    0.000    4.280    2.802
##    .Im3               4.995    0.056   88.571    0.000    4.995    3.786
##    .Im4               4.999    0.057   87.000    0.000    4.999    3.713
##    .Im5               5.036    0.057   87.852    0.000    5.036    3.787
##    .Im11              5.653    0.049  115.303    0.000    5.653    4.944
##    .Im12              5.665    0.049  116.165    0.000    5.665    4.987
##    .Im13              5.448    0.052  105.695    0.000    5.448    4.528
##    .Im6               5.827    0.051  113.792    0.000    5.827    4.857
##    .Im7               5.752    0.052  111.063    0.000    5.752    4.765
##    .Im9               5.075    0.058   87.406    0.000    5.075    3.756
##    .Im1               4.792    0.057   84.290    0.000    4.792    3.600
##    .Im2               4.858    0.055   88.417    0.000    4.858    3.781
##    .Im10              6.100    0.037  162.786    0.000    6.100    6.936
##    .Im14              6.138    0.037  165.853    0.000    6.138    7.093
##    .Im17              5.025    0.053   94.560    0.000    5.025    4.043
##    .Im18              4.595    0.060   76.466    0.000    4.595    3.287
##    .Im16              5.135    0.052   99.194    0.000    5.135    4.270
##    .Im19              5.145    0.048  107.020    0.000    5.145    4.576
##    .SAT_1             5.343    0.043  122.950    0.000    5.343    5.239
##    .SAT_2             5.482    0.043  127.738    0.000    5.482    5.455
##    .SAT_3             5.458    0.050  109.430    0.000    5.458    4.774
##    .COM_A1            4.287    0.061   69.747    0.000    4.287    2.983
##    .COM_A2            3.887    0.069   56.667    0.000    3.887    2.420
##    .COM_A3            3.543    0.070   50.857    0.000    3.543    2.178
##    .COM_A4            3.456    0.074   46.672    0.000    3.456    1.991
##    .C_REP1            4.283    0.031  137.513    0.000    4.283    5.859
##    .C_REP2            4.507    0.027  169.648    0.000    4.507    7.250
##    .C_REP3            4.677    0.024  196.940    0.000    4.677    8.445
##    .C_CR1             2.679    0.084   32.075    0.000    2.679    1.375
##    .C_CR3             3.261    0.088   36.880    0.000    3.261    1.572
##    .C_CR4             2.786    0.085   32.902    0.000    2.786    1.405
##     Shoppng_Exprnc    0.000                               0.000    0.000
##     Store_Decoratn    0.000                               0.000    0.000
##     Luxury_Brands     0.000                               0.000    0.000
##     French_Culture    0.000                               0.000    0.000
##     Prdct_Assrtmnt    0.000                               0.000    0.000
##     Gourmet_Food      0.000                               0.000    0.000
##     Trendiness        0.000                               0.000    0.000
##     Professionalsm    0.000                               0.000    0.000
##    .CS                0.000                               0.000    0.000
##    .AC                0.000                               0.000    0.000
##    .RI                0.000                               0.000    0.000
##    .CI                0.000                               0.000    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .Im20              0.644    0.059   10.840    0.000    0.644    0.288
##    .Im21              0.708    0.056   12.626    0.000    0.708    0.377
##    .Im22              0.557    0.061    9.070    0.000    0.557    0.239
##    .Im3               0.214    0.024    8.796    0.000    0.214    0.123
##    .Im4               0.108    0.024    4.485    0.000    0.108    0.059
##    .Im5               0.747    0.049   15.220    0.000    0.747    0.423
##    .Im11              0.817    0.055   14.817    0.000    0.817    0.625
##    .Im12              0.309    0.040    7.805    0.000    0.309    0.239
##    .Im13              0.390    0.045    8.754    0.000    0.390    0.269
##    .Im6               0.431    0.041   10.534    0.000    0.431    0.300
##    .Im7               0.234    0.041    5.682    0.000    0.234    0.161
##    .Im9               1.199    0.080   15.053    0.000    1.199    0.657
##    .Im1               0.089    0.047    1.918    0.055    0.089    0.050
##    .Im2               0.302    0.041    7.314    0.000    0.302    0.183
##    .Im10              0.116    0.019    6.156    0.000    0.116    0.150
##    .Im14              0.067    0.019    3.618    0.000    0.067    0.090
##    .Im17              0.094    0.045    2.085    0.037    0.094    0.061
##    .Im18              0.523    0.054    9.593    0.000    0.523    0.268
##    .Im16              0.602    0.050   11.943    0.000    0.602    0.416
##    .Im19              0.345    0.043    7.943    0.000    0.345    0.273
##    .SAT_1             0.262    0.034    7.733    0.000    0.262    0.252
##    .SAT_2             0.333    0.033    9.973    0.000    0.333    0.329
##    .SAT_3             0.798    0.056   14.348    0.000    0.798    0.610
##    .COM_A1            0.757    0.058   12.960    0.000    0.757    0.367
##    .COM_A2            0.778    0.065   11.906    0.000    0.778    0.301
##    .COM_A3            0.881    0.070   12.504    0.000    0.881    0.333
##    .COM_A4            0.876    0.075   11.720    0.000    0.876    0.291
##    .C_REP1            0.179    0.016   11.295    0.000    0.179    0.335
##    .C_REP2            0.051    0.010    4.947    0.000    0.051    0.133
##    .C_REP3            0.131    0.009   14.061    0.000    0.131    0.429
##    .C_CR1             1.048    0.113    9.309    0.000    1.048    0.276
##    .C_CR3             1.369    0.130   10.572    0.000    1.369    0.318
##    .C_CR4             1.377    0.122   11.292    0.000    1.377    0.350
##     Shoppng_Exprnc    1.591    0.136   11.660    0.000    1.000    1.000
##     Store_Decoratn    1.526    0.107   14.319    0.000    1.000    1.000
##     Luxury_Brands     0.491    0.067    7.337    0.000    1.000    1.000
##     French_Culture    1.008    0.089   11.380    0.000    1.000    1.000
##     Prdct_Assrtmnt    1.683    0.117   14.435    0.000    1.000    1.000
##     Gourmet_Food      0.657    0.049   13.327    0.000    1.000    1.000
##     Trendiness        1.451    0.104   14.017    0.000    1.000    1.000
##     Professionalsm    0.845    0.087    9.730    0.000    1.000    1.000
##    .CS                0.448    0.047    9.459    0.000    0.576    0.576
##    .AC                0.859    0.086   10.029    0.000    0.657    0.657
##    .RI                0.237    0.022   10.939    0.000    0.667    0.667
##    .CI                2.278    0.208   10.939    0.000    0.829    0.829
standardizedsolution(fit_SEM)
##                     lhs op                 rhs est.std    se       z pvalue
## 1   Shopping_Experience =~                Im20   0.844 0.017  49.384  0.000
## 2   Shopping_Experience =~                Im21   0.789 0.020  39.017  0.000
## 3   Shopping_Experience =~                Im22   0.873 0.016  53.843  0.000
## 4      Store_Decoration =~                 Im3   0.936 0.008 113.829  0.000
## 5      Store_Decoration =~                 Im4   0.970 0.007 137.388  0.000
## 6      Store_Decoration =~                 Im5   0.760 0.019  39.210  0.000
## 7         Luxury_Brands =~                Im11   0.613 0.031  19.843  0.000
## 8         Luxury_Brands =~                Im12   0.872 0.019  46.964  0.000
## 9         Luxury_Brands =~                Im13   0.855 0.019  44.604  0.000
## 10       French_Culture =~                 Im6   0.837 0.018  45.738  0.000
## 11       French_Culture =~                 Im7   0.916 0.016  57.055  0.000
## 12       French_Culture =~                 Im9   0.586 0.032  18.394  0.000
## 13   Product_Assortment =~                 Im1   0.974 0.014  71.821  0.000
## 14   Product_Assortment =~                 Im2   0.904 0.015  61.633  0.000
## 15         Gourmet_Food =~                Im10   0.922 0.014  66.472  0.000
## 16         Gourmet_Food =~                Im14   0.954 0.013  71.602  0.000
## 17           Trendiness =~                Im17   0.969 0.015  64.177  0.000
## 18           Trendiness =~                Im18   0.856 0.018  48.888  0.000
## 19      Professionalism =~                Im16   0.764 0.024  32.323  0.000
## 20      Professionalism =~                Im19   0.853 0.021  40.224  0.000
## 21                   CS =~               SAT_1   0.865 0.020  43.715  0.000
## 22                   CS =~               SAT_2   0.819 0.022  38.089  0.000
## 23                   CS =~               SAT_3   0.624 0.031  20.411  0.000
## 24                   AC =~              COM_A1   0.796 0.019  41.485  0.000
## 25                   AC =~              COM_A2   0.836 0.017  50.270  0.000
## 26                   AC =~              COM_A3   0.817 0.018  46.015  0.000
## 27                   AC =~              COM_A4   0.842 0.016  52.127  0.000
## 28                   RI =~              C_REP1   0.816 0.019  42.492  0.000
## 29                   RI =~              C_REP2   0.931 0.015  62.586  0.000
## 30                   RI =~              C_REP3   0.756 0.021  36.070  0.000
## 31                   CI =~               C_CR1   0.851 0.019  45.339  0.000
## 32                   CI =~               C_CR3   0.826 0.020  41.966  0.000
## 33                   CI =~               C_CR4   0.806 0.021  39.024  0.000
## 34                   CS  ~ Shopping_Experience   0.074 0.054   1.360  0.174
## 35                   CS  ~    Store_Decoration  -0.153 0.060  -2.559  0.011
## 36                   CS  ~       Luxury_Brands  -0.032 0.060  -0.543  0.587
## 37                   CS  ~      French_Culture   0.124 0.057   2.172  0.030
## 38                   CS  ~  Product_Assortment   0.198 0.057   3.484  0.000
## 39                   CS  ~        Gourmet_Food   0.069 0.060   1.149  0.251
## 40                   CS  ~          Trendiness   0.005 0.061   0.089  0.929
## 41                   CS  ~     Professionalism   0.480 0.087   5.527  0.000
## 42                   AC  ~ Shopping_Experience   0.410 0.052   7.916  0.000
## 43                   AC  ~    Store_Decoration  -0.028 0.059  -0.480  0.631
## 44                   AC  ~       Luxury_Brands  -0.118 0.059  -1.988  0.047
## 45                   AC  ~      French_Culture   0.208 0.056   3.689  0.000
## 46                   AC  ~  Product_Assortment   0.116 0.056   2.062  0.039
## 47                   AC  ~        Gourmet_Food   0.012 0.060   0.194  0.846
## 48                   AC  ~          Trendiness  -0.028 0.061  -0.450  0.652
## 49                   AC  ~     Professionalism   0.131 0.084   1.554  0.120
## 50                   RI  ~                  CS   0.318 0.063   5.046  0.000
## 51                   RI  ~                  AC   0.356 0.054   6.640  0.000
## 52                   RI  ~ Shopping_Experience   0.086 0.059   1.441  0.149
## 53                   RI  ~    Store_Decoration   0.021 0.060   0.353  0.724
## 54                   RI  ~       Luxury_Brands   0.092 0.060   1.531  0.126
## 55                   RI  ~      French_Culture  -0.069 0.058  -1.189  0.234
## 56                   RI  ~  Product_Assortment  -0.038 0.056  -0.675  0.500
## 57                   RI  ~        Gourmet_Food   0.057 0.059   0.966  0.334
## 58                   RI  ~          Trendiness  -0.018 0.060  -0.295  0.768
## 59                   RI  ~     Professionalism  -0.058 0.093  -0.621  0.535
## 60                   CI  ~                  CS  -0.190 0.068  -2.773  0.006
## 61                   CI  ~                  AC   0.378 0.058   6.572  0.000
## 62                   CI  ~ Shopping_Experience   0.116 0.066   1.745  0.081
## 63                   CI  ~    Store_Decoration  -0.022 0.067  -0.331  0.741
## 64                   CI  ~       Luxury_Brands   0.085 0.067   1.268  0.205
## 65                   CI  ~      French_Culture  -0.081 0.065  -1.257  0.209
## 66                   CI  ~  Product_Assortment  -0.006 0.063  -0.091  0.927
## 67                   CI  ~        Gourmet_Food  -0.036 0.066  -0.542  0.588
## 68                   CI  ~          Trendiness   0.019 0.067   0.286  0.775
## 69                   CI  ~     Professionalism  -0.099 0.102  -0.966  0.334
## 70                 Im20 ~~                Im20   0.288 0.029   9.996  0.000
## 71                 Im21 ~~                Im21   0.377 0.032  11.818  0.000
## 72                 Im22 ~~                Im22   0.239 0.028   8.439  0.000
## 73                  Im3 ~~                 Im3   0.123 0.015   7.990  0.000
## 74                  Im4 ~~                 Im4   0.059 0.014   4.346  0.000
## 75                  Im5 ~~                 Im5   0.423 0.029  14.344  0.000
## 76                 Im11 ~~                Im11   0.625 0.038  16.516  0.000
## 77                 Im12 ~~                Im12   0.239 0.032   7.381  0.000
## 78                 Im13 ~~                Im13   0.269 0.033   8.218  0.000
## 79                  Im6 ~~                 Im6   0.300 0.031   9.786  0.000
## 80                  Im7 ~~                 Im7   0.161 0.029   5.460  0.000
## 81                  Im9 ~~                 Im9   0.657 0.037  17.603  0.000
## 82                  Im1 ~~                 Im1   0.050 0.026   1.907  0.057
## 83                  Im2 ~~                 Im2   0.183 0.027   6.904  0.000
## 84                 Im10 ~~                Im10   0.150 0.026   5.881  0.000
## 85                 Im14 ~~                Im14   0.090 0.025   3.545  0.000
## 86                 Im17 ~~                Im17   0.061 0.029   2.071  0.038
## 87                 Im18 ~~                Im18   0.268 0.030   8.931  0.000
## 88                 Im16 ~~                Im16   0.416 0.036  11.512  0.000
## 89                 Im19 ~~                Im19   0.273 0.036   7.554  0.000
## 90                SAT_1 ~~               SAT_1   0.252 0.034   7.347  0.000
## 91                SAT_2 ~~               SAT_2   0.329 0.035   9.352  0.000
## 92                SAT_3 ~~               SAT_3   0.610 0.038  15.968  0.000
## 93               COM_A1 ~~              COM_A1   0.367 0.031  12.002  0.000
## 94               COM_A2 ~~              COM_A2   0.301 0.028  10.846  0.000
## 95               COM_A3 ~~              COM_A3   0.333 0.029  11.470  0.000
## 96               COM_A4 ~~              COM_A4   0.291 0.027  10.681  0.000
## 97               C_REP1 ~~              C_REP1   0.335 0.031  10.680  0.000
## 98               C_REP2 ~~              C_REP2   0.133 0.028   4.785  0.000
## 99               C_REP3 ~~              C_REP3   0.429 0.032  13.525  0.000
## 100               C_CR1 ~~               C_CR1   0.276 0.032   8.644  0.000
## 101               C_CR3 ~~               C_CR3   0.318 0.032   9.799  0.000
## 102               C_CR4 ~~               C_CR4   0.350 0.033  10.525  0.000
## 103 Shopping_Experience ~~ Shopping_Experience   1.000 0.000      NA     NA
## 104    Store_Decoration ~~    Store_Decoration   1.000 0.000      NA     NA
## 105       Luxury_Brands ~~       Luxury_Brands   1.000 0.000      NA     NA
## 106      French_Culture ~~      French_Culture   1.000 0.000      NA     NA
## 107  Product_Assortment ~~  Product_Assortment   1.000 0.000      NA     NA
## 108        Gourmet_Food ~~        Gourmet_Food   1.000 0.000      NA     NA
## 109          Trendiness ~~          Trendiness   1.000 0.000      NA     NA
## 110     Professionalism ~~     Professionalism   1.000 0.000      NA     NA
## 111                  CS ~~                  CS   0.576 0.043  13.384  0.000
## 112                  AC ~~                  AC   0.657 0.039  16.940  0.000
## 113                  RI ~~                  RI   0.667 0.039  17.235  0.000
## 114                  CI ~~                  CI   0.829 0.038  21.987  0.000
## 115 Shopping_Experience ~~    Store_Decoration   0.467 0.038  12.450  0.000
## 116 Shopping_Experience ~~       Luxury_Brands   0.418 0.042   9.852  0.000
## 117 Shopping_Experience ~~      French_Culture   0.352 0.043   8.098  0.000
## 118 Shopping_Experience ~~  Product_Assortment   0.447 0.038  11.645  0.000
## 119 Shopping_Experience ~~        Gourmet_Food   0.295 0.043   6.794  0.000
## 120 Shopping_Experience ~~          Trendiness   0.516 0.037  14.088  0.000
## 121 Shopping_Experience ~~     Professionalism   0.476 0.042  11.387  0.000
## 122    Store_Decoration ~~       Luxury_Brands   0.470 0.038  12.426  0.000
## 123    Store_Decoration ~~      French_Culture   0.364 0.041   8.846  0.000
## 124    Store_Decoration ~~  Product_Assortment   0.442 0.036  12.122  0.000
## 125    Store_Decoration ~~        Gourmet_Food   0.417 0.038  11.065  0.000
## 126    Store_Decoration ~~          Trendiness   0.516 0.035  14.808  0.000
## 127    Store_Decoration ~~     Professionalism   0.655 0.032  20.654  0.000
## 128       Luxury_Brands ~~      French_Culture   0.339 0.044   7.645  0.000
## 129       Luxury_Brands ~~  Product_Assortment   0.477 0.038  12.522  0.000
## 130       Luxury_Brands ~~        Gourmet_Food   0.452 0.039  11.615  0.000
## 131       Luxury_Brands ~~          Trendiness   0.565 0.035  16.196  0.000
## 132       Luxury_Brands ~~     Professionalism   0.531 0.040  13.239  0.000
## 133      French_Culture ~~  Product_Assortment   0.248 0.044   5.635  0.000
## 134      French_Culture ~~        Gourmet_Food   0.602 0.032  18.889  0.000
## 135      French_Culture ~~          Trendiness   0.366 0.042   8.717  0.000
## 136      French_Culture ~~     Professionalism   0.392 0.045   8.801  0.000
## 137  Product_Assortment ~~        Gourmet_Food   0.312 0.041   7.561  0.000
## 138  Product_Assortment ~~          Trendiness   0.521 0.034  15.277  0.000
## 139  Product_Assortment ~~     Professionalism   0.602 0.035  17.115  0.000
## 140        Gourmet_Food ~~          Trendiness   0.325 0.041   7.939  0.000
## 141        Gourmet_Food ~~     Professionalism   0.500 0.039  12.715  0.000
## 142          Trendiness ~~     Professionalism   0.602 0.035  17.316  0.000
## 143                  RI ~~                  CI  -0.021 0.052  -0.404  0.686
## 144                Im20 ~1                       3.125 0.103  30.227  0.000
## 145                Im21 ~1                       3.750 0.121  31.050  0.000
## 146                Im22 ~1                       2.802 0.095  29.427  0.000
## 147                 Im3 ~1                       3.786 0.122  30.966  0.000
## 148                 Im4 ~1                       3.713 0.120  30.892  0.000
## 149                 Im5 ~1                       3.787 0.124  30.536  0.000
## 150                Im11 ~1                       4.944 0.157  31.556  0.000
## 151                Im12 ~1                       4.987 0.158  31.554  0.000
## 152                Im13 ~1                       4.528 0.144  31.371  0.000
## 153                 Im6 ~1                       4.857 0.153  31.837  0.000
## 154                 Im7 ~1                       4.765 0.151  31.498  0.000
## 155                 Im9 ~1                       3.756 0.122  30.738  0.000
## 156                 Im1 ~1                       3.600 0.117  30.795  0.000
## 157                 Im2 ~1                       3.781 0.122  30.985  0.000
## 158                Im10 ~1                       6.936 0.213  32.497  0.000
## 159                Im14 ~1                       7.093 0.221  32.061  0.000
## 160                Im17 ~1                       4.043 0.130  31.185  0.000
## 161                Im18 ~1                       3.287 0.109  30.211  0.000
## 162                Im16 ~1                       4.270 0.137  31.197  0.000
## 163                Im19 ~1                       4.576 0.145  31.518  0.000
## 164               SAT_1 ~1                       5.239 0.164  31.993  0.000
## 165               SAT_2 ~1                       5.455 0.171  31.822  0.000
## 166               SAT_3 ~1                       4.774 0.154  31.028  0.000
## 167              COM_A1 ~1                       2.983 0.100  29.691  0.000
## 168              COM_A2 ~1                       2.420 0.085  28.579  0.000
## 169              COM_A3 ~1                       2.178 0.079  27.668  0.000
## 170              COM_A4 ~1                       1.991 0.074  27.034  0.000
## 171              C_REP1 ~1                       5.859 0.179  32.673  0.000
## 172              C_REP2 ~1                       7.250 0.221  32.854  0.000
## 173              C_REP3 ~1                       8.445 0.258  32.684  0.000
## 174               C_CR1 ~1                       1.375 0.060  22.814  0.000
## 175               C_CR3 ~1                       1.572 0.064  24.432  0.000
## 176               C_CR4 ~1                       1.405 0.061  23.127  0.000
## 177 Shopping_Experience ~1                       0.000 0.000      NA     NA
## 178    Store_Decoration ~1                       0.000 0.000      NA     NA
## 179       Luxury_Brands ~1                       0.000 0.000      NA     NA
## 180      French_Culture ~1                       0.000 0.000      NA     NA
## 181  Product_Assortment ~1                       0.000 0.000      NA     NA
## 182        Gourmet_Food ~1                       0.000 0.000      NA     NA
## 183          Trendiness ~1                       0.000 0.000      NA     NA
## 184     Professionalism ~1                       0.000 0.000      NA     NA
## 185                  CS ~1                       0.000 0.000      NA     NA
## 186                  AC ~1                       0.000 0.000      NA     NA
## 187                  RI ~1                       0.000 0.000      NA     NA
## 188                  CI ~1                       0.000 0.000      NA     NA
##     ci.lower ci.upper
## 1      0.810    0.877
## 2      0.749    0.829
## 3      0.841    0.904
## 4      0.920    0.953
## 5      0.956    0.984
## 6      0.722    0.798
## 7      0.552    0.673
## 8      0.836    0.909
## 9      0.817    0.892
## 10     0.801    0.873
## 11     0.885    0.948
## 12     0.523    0.648
## 13     0.948    1.001
## 14     0.875    0.933
## 15     0.895    0.949
## 16     0.928    0.980
## 17     0.940    0.999
## 18     0.821    0.890
## 19     0.718    0.811
## 20     0.811    0.894
## 21     0.826    0.904
## 22     0.777    0.861
## 23     0.564    0.684
## 24     0.758    0.834
## 25     0.803    0.868
## 26     0.782    0.852
## 27     0.811    0.874
## 28     0.778    0.853
## 29     0.902    0.960
## 30     0.715    0.797
## 31     0.814    0.888
## 32     0.787    0.864
## 33     0.766    0.846
## 34    -0.032    0.180
## 35    -0.271   -0.036
## 36    -0.149    0.084
## 37     0.012    0.235
## 38     0.087    0.310
## 39    -0.049    0.187
## 40    -0.115    0.126
## 41     0.310    0.650
## 42     0.308    0.511
## 43    -0.143    0.087
## 44    -0.235   -0.002
## 45     0.097    0.318
## 46     0.006    0.226
## 47    -0.106    0.129
## 48    -0.147    0.092
## 49    -0.034    0.295
## 50     0.195    0.442
## 51     0.251    0.461
## 52    -0.031    0.202
## 53    -0.097    0.140
## 54    -0.026    0.209
## 55    -0.182    0.045
## 56    -0.148    0.072
## 57    -0.059    0.174
## 58    -0.136    0.101
## 59    -0.240    0.124
## 60    -0.324   -0.056
## 61     0.265    0.491
## 62    -0.014    0.245
## 63    -0.154    0.110
## 64    -0.046    0.216
## 65    -0.208    0.046
## 66    -0.129    0.118
## 67    -0.166    0.094
## 68    -0.113    0.151
## 69    -0.299    0.102
## 70     0.232    0.345
## 71     0.315    0.440
## 72     0.183    0.294
## 73     0.093    0.153
## 74     0.033    0.086
## 75     0.365    0.480
## 76     0.551    0.699
## 77     0.176    0.303
## 78     0.205    0.333
## 79     0.240    0.360
## 80     0.103    0.218
## 81     0.584    0.730
## 82    -0.001    0.102
## 83     0.131    0.235
## 84     0.100    0.200
## 85     0.040    0.140
## 86     0.003    0.118
## 87     0.209    0.326
## 88     0.345    0.487
## 89     0.202    0.344
## 90     0.184    0.319
## 91     0.260    0.398
## 92     0.535    0.685
## 93     0.307    0.426
## 94     0.247    0.356
## 95     0.276    0.390
## 96     0.237    0.344
## 97     0.273    0.396
## 98     0.078    0.187
## 99     0.366    0.491
## 100    0.213    0.339
## 101    0.255    0.382
## 102    0.285    0.416
## 103    1.000    1.000
## 104    1.000    1.000
## 105    1.000    1.000
## 106    1.000    1.000
## 107    1.000    1.000
## 108    1.000    1.000
## 109    1.000    1.000
## 110    1.000    1.000
## 111    0.491    0.660
## 112    0.581    0.733
## 113    0.591    0.742
## 114    0.755    0.903
## 115    0.394    0.541
## 116    0.335    0.501
## 117    0.267    0.437
## 118    0.372    0.523
## 119    0.210    0.380
## 120    0.444    0.588
## 121    0.394    0.558
## 122    0.396    0.544
## 123    0.284    0.445
## 124    0.370    0.513
## 125    0.343    0.491
## 126    0.448    0.585
## 127    0.593    0.717
## 128    0.252    0.426
## 129    0.402    0.552
## 130    0.376    0.528
## 131    0.497    0.634
## 132    0.452    0.610
## 133    0.162    0.335
## 134    0.540    0.665
## 135    0.284    0.448
## 136    0.305    0.480
## 137    0.231    0.392
## 138    0.454    0.588
## 139    0.533    0.671
## 140    0.245    0.405
## 141    0.423    0.577
## 142    0.534    0.670
## 143   -0.123    0.081
## 144    2.922    3.327
## 145    3.514    3.987
## 146    2.615    2.988
## 147    3.547    4.026
## 148    3.477    3.948
## 149    3.544    4.030
## 150    4.637    5.251
## 151    4.677    5.296
## 152    4.245    4.811
## 153    4.558    5.156
## 154    4.469    5.062
## 155    3.516    3.995
## 156    3.371    3.829
## 157    3.541    4.020
## 158    6.518    7.355
## 159    6.659    7.526
## 160    3.788    4.297
## 161    3.074    3.501
## 162    4.001    4.538
## 163    4.291    4.860
## 164    4.918    5.560
## 165    5.119    5.791
## 166    4.472    5.075
## 167    2.786    3.180
## 168    2.254    2.585
## 169    2.024    2.332
## 170    1.847    2.135
## 171    5.508    6.211
## 172    6.817    7.682
## 173    7.939    8.952
## 174    1.257    1.493
## 175    1.446    1.699
## 176    1.286    1.524
## 177    0.000    0.000
## 178    0.000    0.000
## 179    0.000    0.000
## 180    0.000    0.000
## 181    0.000    0.000
## 182    0.000    0.000
## 183    0.000    0.000
## 184    0.000    0.000
## 185    0.000    0.000
## 186    0.000    0.000
## 187    0.000    0.000
## 188    0.000    0.000

SEM Visualization

lavaanPlot(model = fit_SEM, node_options = list(shape = "box", fontname = "Helvetica"), edge_options = list(color = "steelblue4"), coefs = TRUE, sig = 0.05, covs = FALSE, digits = 2)
lavaanPlot(model = fit_SEM, node_options = list(shape = "box", fontname = "Helvetica"), edge_options = list(color = "steelblue4"), coefs = TRUE,sig = 0.05, covs = TRUE, digits = 2)